The Black Cats fell a goal down in the 11th minute when Elijah Adebayo converted from close range after a Hatters corner.
Sunderland drew level before half-time, though, with Amad scoring an excellent equaliser following Alex Pritchard’s short free-kick.
The hosts then took the lead midway through the second half when Trai Hume headed home Jack Clarke’s cross.
It means Sunderland will take a lead into Tuesday’s second leg at Kenilworth Road.
Here’s how Phil Smith rated the Sunderland players against Luton:
1. Anthony Patterson - 7
So unfortunate that a brilliant save early on eventually fell to Adebayo to score. Claimed crosses well and actually had little to do in terms of saves. 7 Photo: Martin Swinney
2. Trai Hume - 8
Has been getting into decent positions all season and made it count with an excellent header, guiding the cross from Clarke into the corner at the near post. Defensively strong, too. 8 Photo: Frank Reid
3. Luke O’Nien - 7
Kept his nerve in possession even as Luton pressed, especially as there was never an option to go long. Got through the game well after taking a yellow card when his side were on the ropes early on. 7 Photo: Frank Reid
4. Lynden Gooch - 7
Had one or two nervy moments but impossible to criticise when playing so completely out of position. A big shift on the left of the back three, as Sunderland somehow roll on despite their major injury crisis. 7 Photo: Frank Reid