After wasting a plethora of opportunities to take the lead that their performance merited, Thomas Frank's men were on the verge of suffering an undeserved defeat against their lacklustre opponents, who broke the deadlock late on through Mason Mount's first goal for the club.
However, Mount's strike was the catalyst that the hosts needed to finally find a clinical edge, as Kristoffer Ajer denied Erik ten Hag's troops maximum spoils in remarkable fashion to snatch a potentially priceless point for Brentford in the fight against the drop.
Following a last-minute scare over Kobbie Mainoo, who reportedly sat out training earlier this week due to illness, the England international was cleared to start for Man United, who were also boosted by the returns of Lisandro Martinez, Harry Maguire and Casemiro to the bench.
Meanwhile, Yegor Yarmolyuk benefitted from Christian Norgaard's latest injury to take his place in the Bees' engine room, and it was Frank's relegation battlers who were asking most of the questions in the opening period.
A prolonged spell of pressure just after the 20-minute mark ended with Zanka heading over the bar, just two moments before Ivan Toney got in behind the Red Devils' backline and also beat Andre Onana with a low strike, but the post came to Man United's rescue.
Brentford bark without the bite
Yoane Wissa emulated Zanka with a header over the top just one minute later as Brentford upped the ante, and Man United were given no time to catch their breath, as the hosts forced set piece after set piece.
Another corner situation in the 33rd minute ended in another close shave, as Zanka again won the aerial battle and grazed the crossbar, four minutes before Vitaly Janelt fired just wide when Ten Hag's beleaguered men failed to clear their lines.
A speculative 42nd-minute strike from Marcus Rashford - which was straight down Mark Flekken's throat - was a rare moment of promise in an otherwise abysmal half from the visitors, although Brentford had failed to shine their best shooting boots, firing 14 attempts without success in the opening period.
Ten Hag saw fit to introduce the fit-again Maguire for Raphael Varane during the break, and an end-to-end start to the second period saw both goalkeepers immediately forced into action, as Flekken firstly produced a top stop to keep Rasmus Hojlund out in the 53rd minute.
Just two minutes later, Onana denied Yarmolyuk at full stretch down to his right before also thwarting Keane Lewis-Potter from close range on the follow-up, although he could only stand and watch a wonderful Wissa volley fizz a whisker wide of his post in the 58th minute.
Toney and Lewis-Potter also cleared the crossbar with volleys of their own as Man United hung on by the skin of their teeth, and the Bees nearly paid the price for their wasted openings in the 72nd minute, when substitute Antony curled a fraction wide from outside the box.
Man United's top-four hopes dwindle
The Brazilian's near-miss came seconds before Toney finally found the mark with a volley from Bryan Mbeumo's cross, but the offside flag quickly shot up, and the linesman's call was upheld following a VAR review.
Off-the-ground efforts were still the order of the day as the contest entered its final stages - a Flekken punt forward ended with Mbeumo's half-volley rattling the bar in the 78th minute - and the tie was seemingly destined to end without the net bulging once.
However, Mount - whose debut season at Man United has been forgettable - would spark bedlam in the away end six minutes into second-half injury time, firing into the far corner after Casemiro managed to poke the ball through to the Englishman.
The Red Devils' celebrations were premature, though, as a hitherto profligate Brentford finally put one of their endless chances away in the eighth additional minute, when Ajer was afforded far too much time and space in the centre of the box to convert from Toney's cutback.
Owing to the Scandinavian's quick response, sixth-placed Man United are now eight points adrift of Aston Villa and Tottenham Hotspur as their Champions League hopes dwindle, while 15th-placed Brentford have a five-point cushion over the relegation zone.
The midweek schedule sees Frank's team welcome Brighton & Hove Albion to West London on Wednesday, just over 24 hours before Man United face fallen titans Chelsea at Stamford Bridge.