After a goalless first half, the visitors took the lead six minutes after the restart when Alfie Doughty’s low effort beat goalkeeper Anthony Patterson.
Sunderland reacted well, though, with Mowbray using all five of his available substitutes.
One of those subs then won a penalty as Amad was fouled in the Luton box, before converting from the spot.
Here’s how Phil Smith rated the Sunderland players against Luton:
5. Luke O’Nien - 8
Defended well and fed Clarke excellently with some precise passing. Has done really well at left back, and has a big future in this backline. Deserved to be on the winning side. 8 Photo: Frank Reid
6. Dan Neil - 6
Got away with losing the ball in dangerous areas once or twice but generally moved it well and was very unlucky to hit the post with a brilliant long-range effort in the first half. 6 Photo: Martin Swinney
7. Edouard Michut - 5
Had a really good first half especially out of possession where he made some good challenges and won the ball regularly. Couldn’t reach that level again after the break. 5 Photo: Frank Reid
8. Patrick Roberts - 6
One or two excellent runs early on almost yielded a reward, but in general Luton were very effective at swamping him with a number of players when the ball was played into him. Kept going throughout but ultimately one of his quieter days. 6 Photo: Martin Swinney