Lammers came incredibly close to sending Twente ahead in the early exchanges, with the forward just missing the target with his low effort from a Bart van Rooij cross.
Down the other end, Twente goalkeeper Lars Unnerstall made a wonderful save to prevent his teammate Mees Hilgers from finding the back of his own team's net with a misjudged clearance.
Man United were pressing, though, and the opening goal came in the 35th minute, with Eriksen thumping one into the far corner of the net after the ball fell kindly for the Dane inside the penalty box.
Twente had their moments in the opening half of action, but the hosts were the dominant outfit, with Amad Diallo and Marcus Rashford impressing for the 20-time English champions in the wide positions.
The visitors were always in the game at 1-0, though, and the leveller came in the 68th minute through Lammers, who fired into the back of the net after a mistake from Eriksen.
As expected, Erik ten Hag's side were pushing for a late second in Manchester, but it was not to be, as the Dutch visitors held on for an excellent point in their opening match of the new-look Europa League.
MAN UNITED VS. TWENTE HIGHLIGHTS
Lars Unnerstall save vs. Man United (26th min)
Incredible stop from the Twente goalkeeper to keep out what was supposed to be a clearance from his own teammate Hilgers; Man United almost take the lead at Old Trafford in bizarre circumstances.
Christian Eriksen goal vs. Twente (35th min, Man United 1-0 Twente)
Man United make the breakthrough in the 35th minute of the contest, and it is a wonderful hit from Eriksen, as the Dane picks out the far corner with a thumping effort from inside the penalty box.
Sam Lammers goal vs. Man United (68th min, Man United 1-1 Twente)
Twente level the scores in the 68th minute of the contest, and it is Lammers with the goal.
A mistake from Eriksen allows Lammers to break through, and the forward keeps his cool to fire past Onana, with the Cameroonian leaning to go the opposite way.
MAN OF THE MATCH - MEES HILGERS
Hilgers almost scored an incredible own goal in the first period, but the defender recovered from that moment to put in an excellent display for the Dutch outfit.
The 23-year-old made three tackles and won one aerial duel, while he used the ball brilliantly, finishing with a pass success rate of 92%.
Hilgers was particularly impressive late on when Twente came under pressure, standing strong to earn his side a welcome point in their Europa League league-phase opener.
MAN UNITED VS. TWENTE MATCH STATS
Possession: Man United 56%-44% Twente
Shots: Man United 17-8 Twente
Shots on target: Man United 5-3 Twente
Corners: Man United 10-5 Twente
Fouls: Man United 4-7 Twente
WHAT NEXT?
Man United's attention will now switch back to the Premier League, as they prepare to welcome Tottenham Hotspur to Old Trafford on Sunday afternoon.
Twente, meanwhile, will be bidding to build on the strong result when they resume their Eredivisie campaign with a home fixture against NAC Breda on Sunday afternoon.