But it could have been a very different story had Alex striker Dean Ashton scored from the penalty spot after Carl Robinson had brought down Steve Jones.
The 10-goal forward blazed his spot-kick over the crossbar to extend Crewe's winless run to five games.
United had Mike Whitlow sent off in the 28th minute after he was adjudged to have clipped the heels of Craig Hignett as the midfielder ran towards goal.
The veteran defender was shown a straight red card, despite having earlier been booked for a late challenge on Ashton.
The visitors made a lively start and came close to breaking the deadlock after four minutes as Phil Jagielka's cross was headed inches wide by Montgomery.
Clayton Ince pulled off a fine save to deny youngster Jonathan Forte shortly afterwards.
The Railwaymen hit back as David Vaughan tested Paddy Kenny with a 20-yard shot, but the Blades were dominating, and Ince was called into action again to keep out a fierce drive from Michael Tonge.
United were dealt a blow as Whitlow was given his marching orders but reorganised and looked more likely to score as the half drew to a close.
Forte, making his full debut at 17, forced another good stop from Ince with a well-struck shot from a tight angle, with the Trinidad & Tobago stopper palming away the follow-up from Montgomery.
The visitors continued to push forward at the start of the second half, with Forte again causing problems for the Alex defence.
His 20-yard shot flicked off David Wright and looped goalwards, forcing Ince to parry, with defender Stephen Foster clearing for a corner.
Ashley Ward was then denied the opener as his header was hacked off the line by Hignett.
Dario Gradi's side should have gone ahead in the 51st minute. A shot from substitute Billy Jones was pushed away by Kenny, and Foster put the rebound wide from close in.
Tonge and Parkinson tried their luck from distance before Ashton missed from the penalty spot.
They were made to pay with five minutes remaining.
Substitute Peter Ndlovu raced down the right and cut the ball back for Forte. His shot was blocked by Ince but the ball bounced up kindly for Montgomery to loop a header over Kenny Lunt on the line.