A Kevin Nicholls penalty and a 15th of the season from Ahmet Brkovic gave Town a flattering 2-0 lead before Michael Proctor replied for the visitors before the break.
And after pressure from the Robins in the second period it took a late strike from Peter Holmes to make the game safe for the League One leaders.
The Hatters struggled to get into the match during the opening exchanges as Swindon controlled possession in the first 10 minutes.
But the hosts took the lead somewhat against the run of play in the 12th minute when Matthew Heywood fouled Brkovic in the area and skipper Nicholls confidently stroked the ball past visiting goalkeeper Rhys Evans for his 11th goal of the season.
Swindon almost replied when Grant Smith volleyed at Marlon Beresford from 18 yards after Sam Igoe's chipped pass.
But Town threatened themselves in the 17th minute when Steve Howard's flick-on from Keith Keane's cross almost let in Paul Underwood only for the winger to be denied by a last-ditch challenge from Jerel Ifil.
Luton doubled their advantage in the 26th minute when Sol Davis fired in an effort from the edge of the area that bounced off Rowan Vine's boot directly into the path of Brkovic who calmly slotted past Evans from six yards.
It looked plain sailing for the table-toppers, but the scoreline was given a truer reflection of the balance of play in the 34th minute when Swindon made it 2-1.
Grant Smith's corner from the right appeared to beat a posse of players in the goalmouth and go straight into the net, although the Robins' Rotherham loan man Proctor may have got a touch.
The visitors pushed again and another Smith corner two minutes later saw a deflected Sam Parkin header tipped over the bar by Beresford.
Swindon's 4-5-1 formation continued to frustrate the Hatters after the break as the hosts again had the better of possession.
But a Luton break in the 50th minute saw Underwood cleverly dummy a Nicholls pass out wide allowing Davis to burst towards the edge of the area only to be crudely chopped down by Proctor who was booked for the foul.
Nicholls stepped up and drilled the 20-yard free-kick just past the far post.
Howard then shot weakly at Evans from the edge of the area in the 54th minute before Luton came close to grabbing a third three minutes later when Underwood's cross was twice headed by Holmes before Underwood himself collected the loose ball, sprinted into the area and crashed a volley against the bar from 15 yards.
Swindon came back into things after that and almost equalised when substitute Christian Roberts hit an eight-yard volley just past the post after Igoe's cross nine minutes from time.
But Luton fans' nerves were eased three minutes later when a deep cross by Nicholls was dropped by Evans and Holmes took delight in placing the ball into an empty net from 12 yards for his second goal in successive home games.