After a scrappy start, Town began to grow into the game and soon rattled the visitors as Natalie Richardson's ambitious long-range free-kick smashed the crossbar.
Thomas was proving to be a constant threat down the Ipswich right but her dangerous split cross couldn't be turned in by a blue shirt.
Stevenage could have been awarded a penalty moments later but the referee ruled Laken Duchar-Clark's challenge as shoulder to shoulder, incensing the visitors' bench.
Stevenage started the second-half the brighter, with Amy Makewell proving to be a thorn in the side for Town.
However, Ipswich took the lead after a rapid counter-attack as Thomas galloped through the middle of the field before firing low into the corner from the edge of the area.
The game would then have a dramatic finish with Town having two glorious chances to seal victory. First, Thomas fed the ball into the path of Sophie Welton but the winger blazed over from inside the area.
Next, Stevenage themselves sprung a counter-attack and looked certain to score, only for Makewell to be denied by the post.
Town did have the ball in the net immediately after as Zoe Cossey burst through the Stevenage defence but the wingers' effort was touched in by the offside Welton.
Stevenage then delivered the ultimate sucker punch with an equaliser with just seconds remaining. Makewell's free-kick rebounded off the bar and fell to Ashleigh Deacon, who turned home at the back-post to send the game to extra-time. Cossey nearly gave Ipswich the lead for a second time but her driven effort was excellently tipped on to the bar by Hannah Vanderluis.
However, the Hertfordshire side were buoyed by their spirited fightback and were soon in front. Ellie Searle found herself in behind and found the back of the net after an uncharacteristic blunder from Town goalkeeper Sian Fagg.
The Blues would have one final opportunity to send the game to penalties but Welton fired straight at Vanderluis after being released one-on-one by skipper Kerry Stimson. Despite Town pressure, Stevenage would eventually hold on to book their place in the quarter-finals.
Town XI: Fagg (GK), Ball (Cooper), Petley, Duchar-Clark, Pannifer, Stimson (c), Craddock (Wakefield), Welton, Richardson (Cossey), Thomas, Wayne.
Unused: Clarke (GK), Cage.