Ben Wilmot gave the visitors the lead early on as he headed past Tiernan Parker, before Andronicos Georgiou curled home in fine style just after the restart to double their lead.
Substitute Michael Shamalo got on the score-sheet late for the Hatters as he curled past Nathan Harness, but it proved to be nothing more than a consolation goal.
Stevenage dominated the early proceedings as Joe Leslie fired just wide after two minutes before they broke the deadlock in the sixth minute.
A corner from the far side was swung in and defender Wilmot rose at the back stick to power a header beyond Parker and into the net to give the visitors an early lead.
Arthur Read went close to bringing the game level 15 minutes in when his curling effort from the edge of the box was well kept out by Nathan Harness who dived to his right and got two strong palms on it to push it away from goal.
Scott Belgrove was then at the heart of the action for the rest of the half as he saw his close range effort from a Corey Panter knock-down drift wide just before the 30 minute mark. With six minutes of the half remaining his strike from the edge of the box deflected towards the bottom left corner, but Harness was there once again to pull off a decent save.
With the Hatters pushing for a goal just before the break, Belgrove's header was pushed off the line by Harness, before the visitors went on the counter attack and the Town midfielder brought down Marcus Gouldbourne – picking up a yellow card in the process.
The hosts doubled their lead shortly after the restart, when Andronicos Georgiou pounced on a loose ball before curling it into the bottom left corner from 20 yards in emphatic fashion in the 49th minute.
Neufville once again tried to get his side back into the game, but his effort after 65 minutes drifted well over the crossbar.
The visitors should have found themselves 3-0 up when Joe Leslie met Georgiou's cross at the back stick three yards out, but his effort flew over the bar with 15 minutes left to go.
Parker pulled off a fantastic save in the final minute of normal time as he dived to his left to keep out Georgiou's fine strike.
It kept the Town's faint hopes of getting something from the game alive and a minute later Idiakez's side had halved the deficit as Shamalo tucked away Joe Mead's superb pass to set up a tense ending, but the visitors saw out the result to progress to the next round.
TOWN: Parker, James, Mead ©, Panter, Byron, Sorunke (Richardson 60), Peck, Read, Belgrove (Jones 76), Tomlinson (Shamalo 61), Neufville.
Subs: Ellis, Shamalo, Beckwith, Richardson, Jones.
Goal: Shamalo 90+1
Yellows: Belgrove
STEVENAGE: Harness, Sackey-Mensah, Wilmot, O'Keefe ©, Omar, Taylor-Cole, Gouldbourne, Lontton, Draper, Georgiou, Leslie.
Subs: Fernandez, Payne, Smyth, Field, Moriarty.
Goals: Wilmot 6, Georgiou 49
Yellows: Wilmot
Referee: N Morrison
ATTENDANCE: 392
