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:
9. Abdoullah Ba - 5
Played one brilliant pass for Michut that almost yielded a goal and carried the ball nicely at times, but generally struggled to impose himself against a physically powerful and very well-organised opponent. 5 Photo: Frank Reid
10. Jack Clarke - 7
Looked a little leggy given his recent workload in the first half but was excellent after the break and at the heart of almost all of Sunderland’s best moves. Could have nicked a winner late on and he would have deserved it. 7 Photo: Martin Swinney
11. Joe Gelhardt - 5
Ran relentlessly but didn’t trouble the Luton back three a great deal, and missed his one big chance to level the scores shortly before coming off. 5 Photo: Frank Reid
12. Amad (sub) - 7
Lifted his side, showing for the ball and looking to make things happen. Won the penalty and dispatched it emphatically. 7 Photo: Frank Reid