The point means United could not follow up their victory against arch-rivals Liverpool last week which ensured Champions League qualification for next season.
The Red Devils created little pressure in the final third, and instead ot was the home team that showed flashed upfront but goalkeeper Lammens came up trumps.
Chemsdine Talbi went close after just five minutes, but curled his effort with Lammens sprawling.
The Belgian shot-stopper then had had to be quick off his line to deny Brian Brobbey, who broke clear after evading the attention of returning defender Lisandro Martinez.
Sunderland then had a strong penalty claim turned down in the 19th minute when Enzo Le Fee's shot appeared to hit the hand of Kobbie Mainoo, the referee waived play-on.
United only showed intent to get at the hosts late in the second half as Matheus Cunha had an effort saved, but it the end a point was all they barely deserved.
Pep Guardiola has said he will make a late decision on the fitness of Rodri and Abdukodir Khusanov ahead of Manchester City's crucial Premier League clash with Crystal Palace on Wednesday night.
Former Manchester United defender Jaap Stam has backed the idea of a swap deal involving Marcus Rashford and Rafael Leao as uncertainty continues to surround Rashford's long-term future.
Mathys Tel endured a dramatic night in North London as his brilliant goal was cancelled out by a costly mistake in Tottenham Hotspur's 1-1 Premier League draw against Leeds United.
Harry Maguire believes Leny Yoro and Ayden Heaven are set to develop into Manchester United's long-term centre-back pairing, insisting both young defenders have amazing potential.
