Arguably, United should have had a chance to make it a two-goal cushion from the penalty spot.
For the second time in three weeks, a ball struck Emmanuel Fernandez's arm in the penalty box - this time after a long throw - but again no penalty was awarded.
A relived Rangers charged up the other end in search of a leveller.
Substitute Youssef Chermiti had a golden opportunity to equalise but, from the middle of the six yard box and unmarked, he glanced his header wide.
Instead, the opportunity came with the final kick of the game as Max Aarons was fouled in the box by Will Ferry. Up stepped Bajrami, and he curled it into the top corner.
The draw makes it just four wins from Danny Rohl's first 10 games in charge of Rangers, with Celtic and Hearts nine points ahead at the top of the league.
United, meanwhile, stay eighth, five point adrift of the top six.