Alex Oxlade-Chamberlain wasted an excellent chance from 10-yards out to open the scoring midway through the first-half, with Luis Suárez failing to make the most of the Spanish side's best opportunity prior to the break, heading Daniel Alves' cross wide.
After Petr Čech denied Neymar with a strong stop early into the second-half, Arsenal showed more of a willingness to attack after the break - Marc-André ter Stegen's save the only thing denying Olivier Giroud's header from finding the bottom corner.
But Messi finally scored the first goal of the game as a 16-second counter-attack saw Neymar tee the ball up for him to finish and put his side in cruise control at the Emirates Stadium.
After substitute Mathieu Flamini brought down Messi inside the box just moments after coming on, the No.10 sent Čech the wrong way to leave the reigning European champions clear favourites to reach the next round.
Arsenal the better side early on despite strong Barcelona line-up
Arsené Wenger returned to a more familiar full-strength line-up after resting several key players in the 0-0 FA Cup fourth-round draw at home to Hull City on Saturday afternoon.
First-choice goalkeeper Čech returned in goal, with Francis Coquelin, Aaron Ramsey, Mesut Özil, Alexis Sánchez and Olivier Giroud all coming back into the starting eleven - whilst Oxlade-Chamberlain made a surprise start on the right of an attacking three behind Giroud.
The Gunners hosted a star-studded Barcelona side that boasted the scintillating trident of Messi, Suárez and Neymar - who came into the game having scored 30 more goals than Arsenal's entire team in all competitions this season, with an impressive 91.
They spearheaded a strong side, the same one which started in the final of the competition last term and had not been beaten since October 4, a run which had seen the Catalan giants go 32 games unbeaten.
Luis Enrique made four changes from the team that beat Las Palmas in La Liga days earlier, bringing in Marc-André ter Stegen for Claudio Bravo in between the sticks, with Gerard Piqué, Sergio Busquets and Ivan Rakitić amongst the others to return.
Despite a relatively cagey start, it didn't take long for Barcelona's main threats to take centre-stage - Neymar being hacked down by Ramsey down the left-side after a bursting run, although Daniel Alves could only flick Messi's delivery wide with a header.
But despite that, Arsenal actually edged the opening stages - defending with a low block and denying their opponents time and space on the ball, whilst posing a threat on the counter-attack.
Their best early moment came in the eighth minute after Ramsey's backheel into the path of Özil on the left flank, but after returning the ball to the unmarked Welshman in the box - he was quickly closed down by Javier Mascherano, who blocked his tame effort.
Hunting the away side down in packs, Wenger's side managed to keep their opponents largely quiet - Čech being asked to come and collect Alves' poor cross the only action he was called into inside 20 minutes.
Alves had greater success a few minutes later as he found Suárez in the inside channel, but the Uruguayan striker was unable to get past Laurent Koscielny after wriggling through a number of challenges whilst attempting to break into the box.
Arsenal could, and should, have gone ahead after 22 minutes as they started to enjoy a good spell and the tempo picked up. After Özil found space to pick out Bellerín, whose side-footed shot was easily blocked, the ball fortuitously fell towards Oxlade-Chamberlain ten yards from goal - but with Ter Stegen off his line and on the ground, he could only fire a scuffed strike straight at the goalkeeper when just about any elevation would have seen him open the scoring.
Visitors go close in the latter stages of the first-half
The Gunners had plenty of promising spells in possession, but were unable to apply the cutting edge they needed and Barcelona began to turn the screw in response, although they too often lacked the precision with their final pass.
Messi couldn't keep his header down from Neymar's teasing cross just past the half-hour despite rising above Nacho Monreal, before Suárez overran the ball on his way into the six-yard box after Neymar slipped him in with a through ball.
With Barcelona gradually starting to get into full flow, Koscielny could only bring Messi down on the edge of the box after a trademark solo run, but the Argentine couldn't squeeze his attempt at an under-the-wall free-kick through the number of bodies.
After Oxlade-Chamberlain spent a brief spell receiving treatment after a heavy coming-together with Mascherano, the Englishman was guilty of over-running the ball with supporting options in the centre of the pitch and Barcelona continued to apply pressure.
Neymar's jinking run saw him evade a number of challenges after cutting inside from the left. He spotted Rakitić out on the right but Messi's strike in the box was blocked.
But the best chances for Luis Enrique's men came right before the break, Suárez finding the ball at his feet after Messi's flick - but rather than shoot, his pull-back across the face of goal was well cleared.
Suárez had another golden opportunity when Alves' volleyed cross from Busquets' ball invited the striker to flick a header back towards the far post, but he couldn't guide it inside the woodwork as the ball fell wide to deny him a 42nd goal of the season and Čech went without having to make a first-half save.
End-to-end second-half sees both sides create chances
Despite having to uncharacteristically sit back, defend and watch their opponents dominate the ball - the Gunners came out of the first-half with an abundance of positives and entered the second 45 minutes with confidence.
Nevertheless, they had to keep their concentration as Neymar was fed in to a promising position inside the box by Iniesta, but as he looked to slip a shot past Čech, the goalkeeper spread himself well to deny the Brazilian.
Wenger was then forced into an early substitution as Theo Walcott, who was perhaps surprisingly benched in favour of his withdrawn compatriot, replaced Oxlade-Chamberlain - who had failed to shake off the knock he picked up in the aforementioned heavy tackle with Mascherano.
Barcelona continued to try and force the away goal as their front-three posed countless problems, Neymar once again getting into the box, before a one-two with Messi - but rather than shoot, he tried to cross and couldn't pick out a teammate.
They were perhaps fortunate not to see Jordi Alba sent off in the 54th minute, after he reacted angrily to Giroud catching him with a accidental flailing arm, but though the left-back raised his head towards the Frenchman, no action was taken by the referee.
Despite the all-too-frequent stoppages, the tempo of the game invited both sides to attack more often and Arsenal almost took advantage, Giroud first denied by Mascherano's interception before Ter Stegen produced an excellent low left-handed save to push his header destined for the bottom corner away.
Messi at the double to put Barcelona in control
A breathless second-half saw Barcelona go close again as Messi's shoulder feint saw him shape to shoot 12-yards out - but Per Mertesacker threw himself in front of the ball before Suárez dragged his drilled shot just a few inches wide of the far post from an angle.
But the deadlock was finally broken in the 71st minute after a lightning quick-counter caught the Arsenal defence out of place and Neymar ran through the middle onto Suárez's pass before squaring to Messi in the area, and the Argentine simply controlled and placed his left-footed effort into the bottom corner.
That allowed them to settle the game and take control, their patience in the ball frustrating the home supporters who were desperate for their side to claw back an equaliser.
They were almost dealt a devastating blow as Suárez struck the base of the near post from close-range after Messi's set-up, so quickly after Neymar had seen a shot blocked across the opposite side of the box, but much to the relief of the fans the ball was scrambled clear.
And in a pulsating finale, Ramsey almost levelled from substitute Danny Welbeck's headed knockdown, but Busquets was just able to poke it out of the midfielder's reach to allow to Ter Stegen clear and keep his side in-front.
The game was finally put out of reach when Flamini, only brought on in place of Coquelin a minute earlier, brought down Messi in the box following Mertesacker's miscontrol and with one goal already, the World Player of the Year confidently despatched from the spot to make it 2-0.
Barcelona almost made it a three-goal advantage in stoppage-time but Čech's reflexes saw him keep Neymar's header out, although the Gunners couldn't reduce the arrears to leave themselves needing to mount a monumental comeback in the return leg next month.