United boss Carrick was seconds away from heading into the international break on the back of another devastating defeat that would surely have led to his dismissal.
But Carrick’s side struck twice in the six minutes of additional time through Joshua Zirkzee and Patrick Dorgu to leave their manager’s future still uncertain.
United are in the relegation zone and have not won in their last eight Premier League matches, so it is far from sure that Carrick will survive the axe.
He had earned a stay of execution when United’s hierarchy allowed him to remain in charge despite last weekend’s loss at Chelsea.
United responded by ending a nine-match winless run in all competitions as they came from two goals down to beat Roma 3-2 in the Champions League on Tuesday.
But United, who have only three wins in 15 games this season, were back in turmoil when they fell 2-0 down against Villa in the first four minutes at Old Trafford.
Alysson struck in the first minute with a shot that squirmed under United keeper Senne Lammens.
Nicolas Jackson slotted in the second three minutes later after a weak save from Lammens pushed the ball into his path.
Bruno Fernandes got one back with a 16th-minute penalty after Boubacar Kamara fouled Matheus Cunha before former Villa midfielder Youri Tielemans slammed in United’s 44th-minute equaliser.
Villa’s John McGinn converted a 48th-minute penalty following Diogo Dalot’s clumsy challenge on Alysson.
Leon Goretzka poked in Villa’s fourth from close-range after United’s Leny Yoro headed against his own crossbar in the 53rd minute.
Yet Zirkzee’s stoppage-time finish gave United hope and Dorgu headed their dramatic equaliser in the final seconds after a weak punch from Villa keeper Zion Suzuki.
Carrick wore a bewildered look on his face as he walked down the tunnel towards an uncertain future, with United next in action at Liverpool on November 22.
“Making it hard for ourselves. The goals are coming from individual mistakes. We need to tighten up,” Carrick said.
“But I thought we were the better team. If we get one, then anything can happen. We never gave up and never stopped believing.”
Chelsea arrived at the Stadium of Light hoping to bolster their bid to qualify for the Champions League via a top-four finish.
Instead, they departed red-faced after an embarrassing loss to third-bottom Sunderland, who celebrated a league victory for only the second time this season.
Fifth-placed Chelsea have already been beaten four times in 10 league matches under former Real Madrid boss Xabi Alonso.
Chelsea were rocked in the 36th minute when Enzo Le Fee converted a penalty after Brian Brobbey was up-ended by Levi Colwill.
“We were not creating enough chances in the final third against the low block. We were not controlling and sustaining the attack there,” Alonso said.
Liverpool got back on track with a 3-1 win at lowly Crystal Palace.
The Reds had been beaten by Arsenal and Fenerbahce before arriving in south London, sparking pointed questions about Andoni Iraola in the former Bournemouth manager’s first season in charge.
But Alexis Mac Allister headed Liverpool in front from Dominik Szoboszlai’s cross after 40 minutes.
Florian Wirtz bagged Liverpool’s second with a composed finish from Bradley Barcola’s cross in the 54th minute.
Alexander Isak finished off Wirtz’s pass in the 89th minute to wrap up fourth-placed Liverpool’s much-needed victory after a two-game losing streak.
Eddie Nketiah’s stoppage-time penalty after a foul by Milos Kerkez was no consolation for bottom of the table Palace.
Brighton extended their superb start to the season, climbing to second place with a 2-0 win against Brentford at the Amex Stadium.
Diego Gomez put Brighton ahead early on before Brentford striker Igor Thiago was sent off.
Pascal Gross doubled Brighton’s advantage with a 42nd minute penalty.
The Seagulls’ sixth win in their last seven league games lifted them within five points of champions Arsenal, who beat Hull on Saturday.
Bournemouth striker Evanilson maintained his hot streak with a goal in a 2-1 win at Ipswich. (AFP)