The last FA Cup meeting between these two sides came back in December 2003, when the Swans came away with a 2-1 victory, Kevin Nugent and Lee Trundle with the goals.
Stevenage have progressed from four of their six FA Cup third round ties, with this their first since 2017/18 when they went out after a replay against Reading.
This was Swansea's first FA Cup match against League Two opposition since February 2018, when they won 8-1 against Notts County in a fourth round replay.
Wayne Routledge opened the scoring after meeting Yan Dhanda's cross.
The Swans were up at the break and made it two early on in the second half with the swede Viktor Gyökeres finding the net.
Stevenage could've made it a nervy final ten minutes for Swansea, as Luke Norris stepped up to take a penalty, but a fantastic stop from Freddie Woodman prevented that, as the Swans progress by two goals to nil.
Story of the match
The first chance of the game, ended up being the first goal of the game as the Swans took the lead inside seven minutes. Gyökeres found Dhanda out wide, he took his time and found one of the smallest men on the field Routledge, who headed into the back of Stevenage's net. Routledge made a touching tribute to former Swansea employee Nick Eames, who passed away earlier this week.
Swansea almost made it two a minute later, as the ball scuffled inside the area and felt to the feet of Scotsman George Byers, who dragged his shot low into the hands of Stevenage goalkeeper Jamie Cumming.
The first 20 minutes were dominated by the Championship side, as they really should've made it two on this occasion as Jordan Garrick with electric pace down the right hand side, had a simple pass on for Gyökeres to tap it home, but his pass was poor and behind the Swede.
With the Swans dominating the first section of the first-half, Stevenage turned a new leaf and started to play football. The first real chance for the Boro fell to forward Danny Newton inside the area as his strike curled over the bar, Newton was disappointed in himself as he should've done better.
Stevenage were having all the possession leading up to half time, Charlie Carter came close with a header falling inches wide, whilst Ben Coker's free-kick scraped past the post, in which swans goalkeeper Freddie Woodman had covered.
the Boro tried everything to find the net before the interval, but Swansea went into the break with a one goal lead.
The first chance of the second half, should've made the game level. A corner whipped in from Arthur Read, central defender Scott Cuthbert leaped high for the header towards to goal to find the head of his fellow defender Terence Vancooten, who couldn't find enough power on the header to get the ball past Woodman.
Swansea started to turn it on again, after looking lost before half-time. Five minutes into the second-half and they ruined Stevenage's day. Garrick out wide looped in a fantastic ball, which fell perfectly to the head of Gyökeres, in which was his first goal in a swans shirt.
Stevenage tried their hardest to find a way back into the game, but Woodman just wasn't allowing it. Luke Norris had a diving header saved from inside the six-yard box.
In the final ten minutes, a tug of the shirt from Ryan Manning on substitute Romain Vincelot gave Stevenage a lifeline to make it a nervy end for the Swans.
Another substitute Norris stepped up to take the penalty. A decent attempt was brilliantly saved from Woodman to keep Swansea at bay.
It just wasn't Stevenage's day, on another they may have seen themselves through, but it's Swansea City who progress to the next round of the cup.
Man of the match
No other option other than the Swansea goalkeeper Freddie Woodman. Today, he deserves all the plaudits making numerous stops, a brilliant penalty save, and was big and strong when everything came up to face him. A deserved man of the match.