Two goals from Danny Welbeck on his return from injury were followed up by three from Theo Walcott.
The game was practically done by half time, as Arsene Wenger's men were already three goals ahead at the interval.
The Frenchman was serving the first of his four-game touchline ban and was watching on from the stands.
Arsenal started brightly in the early stages, thrashing out three chances in the first ten minutes. Alex Oxlade-Chamberlain, Lucas Perez and Jeff Reine-Adelaide were all causing problems but it was Welbeck who opened the scoring after 15 minutes.
Perez received the ball just outside the penalty area and neatly flicked the ball through to Welbeck, who calmly poked the ball over an oncoming Harry Lewis.
The 26-year-old was on the scoresheet again just six minutes later, when Oxlade-Chamberlain played a fantastic long ball over the top and Welbeck brought it down before slipping it through the legs of Lewis.
The Saints almost fired back instantly, Ospina rushing out of his goal only to be beaten to the ball by Shane Long, whos cross was headed into the path of Hojbjerg, who could only volley over.
Long was again involved moments later. The Irishman was played in behind by Cuco Martina and his half-volley was smartly kept out by David Ospina.
Shkodran Mustafi nearly put the tie to bed after just 30 minutes, as he glanced a header wide from six yards out from an inswinging free-kick.
However, Walcott ensured the game was wrapped up by half time, as he finished a lovely Arsenal move with ten minutes of the first period remaining. Perez again played through Welbeck, who pulled the ball back across goal for Walcott to tap home.
Anything you can do, I can do better
There was no letting up from the Gunners at the start of the second half. Welbeck and Maitland-Niles both had shots wide of the mark, while Hector Bellerin tested Lewis early on.
Josh Sims tried in vain to try and get his side back into the tie with some positive runs down the left-hand side but after a quiet period, Arsenal netted their four in stylish fashion.
Oxlade-Chamberlain jinked his way down the right-hand side and played in substitute Alexis Sanchez, who pulled the ball back to the penalty spot from the by-line, allowing Walcott to tap home from 12 yards.
Sanchez nearly set up a second two minutes later when his one-two with Kieran Gibbs fell just short after he attempted to play the left-back in behind with a backheel.
Southampton kept going nevertheless. Sam McQueen had an effort blocked before firing wide minutes later.
Claude Puel's men were attacking once again when Arsenal sprung a counter attack. Gibbs again finding Sanchez, who took two players out of the game by squaring the ball for Walcott, who once again only needed one touch to finish.
Wenger's men ran out emphatic winners.
Despite the obvious stand out performances from Welbeck and Walcott, the young guns shone for both sides.
Arsenal's Maitland-Niles and Reine-Adelaide were both fantastic throughout the match and have perhaps done enough to force their way into the first-team on a more regular basis.
Also, to add to his cameo role during the week, Southampton's Sims was also the best player for the home side. The youngster looks like he is repaying Puel's faith after he signed a new contract this week.