Casemiro and substitute Benjamin Sesko scored United's other goals in a result that tightened United's grip on third place in the standings on 54 points after 30 games, while fourth-placed Villa have 51 points.
United, who were playing their first game in 11 days, were keen for a victory after interim boss Michael Carrick's seven-game unbeaten run ended in a 2-1 loss to 10-man Newcastle United on March 4.
With 22 points from a possible 27 under Carrick's leadership, this morning's win will invariably lead to more questions about his chances of becoming United's permanent manager.
"It was a slightly different game today going into it being so close in the league and making the small jump we could if we won, off the back of a defeat," Carrick said. "It was a small test and we haven’t really had that.
"The boys dealt with it really well, had a really good week and applied themselves well. Everything is so so encouraging. We’ve got a lot to play for."
The game looked headed for a draw before Cunha scored from the left side of the box in the 71st minute, with Fernandes recording his 100th assist for the club across all competitions.
With eight games left, Fernandes is also four shy of matching the Premier League record for assists in one season of 20 held by Thierry Henry (2002-03) and Kevin de Bruyne (2019-20).
"He's had so many big moments, to score a goal, to make a pass," Carrick said of Fernandes. "Training or games, he’s always there. Really good with the corner. The pass to Matheus was fantastic to cap off a good performance all around."
Sesko put the game out of reach 10 minutes later with a blistering shot from the centre of the box.
Casemiro put United ahead in the 53rd minute with a glancing header from a Fernandes corner.
Ross Barkley levelled in the 64th with a shot from the middle of the area. After a lengthy VAR check to determine whether the ball glanced off an offside Amadou Onana, the goal stood.
A better second half salvaged a decent game after a dreary first 45 minutes plagued by errant passes and slipping players.
United did have some decent first-half chances including a header from Amad Diallo that goalkeeper Emiliano Martinez stretched backwards to keep out of the net. Diogo Dalot had a terrific chance inside the box that he launched over the bar.
Leny Yoro's block spoiled one of Villa's best chances and led to a long-range shot from Fernandes at the other end that he sent just wide of the post.
Villa have not won a league game since February 11, and an animated manager Unai Emery spent much of the afternoon stomping his feet and waving his arms in frustration.
"How we are progressing after we did fantastic months in October, November, December, the last two months we are struggling," Emery said. "It's Manchester United here at home, and they are now in good moment. And of course, they played a very intelligent match.
"I think we can get better. The result at the end was a bad result for us. We must accept it and try to analyse now to get better for the next challenge."
Spurs snatch point against Liverpool
Liverpool missed the chance to return to the Premier League's top four as Tottenham Hotspur striker Richarlison struck a late equaliser in a 1-1 draw at Anfield this morning.
Dominik Szoboszlai's 18th-minute free kick had looked like securing the points for Liverpool but the hosts were unconvincing and could have no complaints as Tottenham ended a five-match losing streak in the league.
Former Everton striker Richarlison was a constant threat for the visitors and fired a low shot past Alisson in the 90th minute to secure a vital point in his side's relegation battle.
Liverpool, who were booed at the end, moved above Chelsea into fifth place with 49 points from 30 games while Tottenham's first point in six Premier League games left them 16th with 30, one point above the relegation zone.
"It's a huge disappointment," Szoboszlai said. "Again in the last minute, I don't know how many times it's happened this season, we have to wake up."
Tottenham are now 12 league games without a win, their worst run since 1935, but the visiting fans were celebrating at the end after a gutsy display that offered some hope that they can avoid a first relegation since 1977.
Interim manager Igor Tudor still has no victories in five games since taking over from Thomas Frank, but his side were unrecognisable from recent chaotic displays, albeit helped by a lacklustre Liverpool who conceded yet another late goal to drop more points in their quest for Champions League qualification.
"It was good. Proud of the guys. They stayed in the game, and we deserve a point," Tudor told Sky Sports.
"The team spirit was amazing. This was the main demand. The quality is there. Every point is crucial, we are just going game by game. We hope today is a turning point."
Despite their many issues, including missing suspended defender Micky van de Ven and captain Cristian Romero because of concussion, Tottenham started brightly as they looked for an unlikely first league win at Anfield since 2011.
But it looked like being a familiar tale of woe as Szoboszlai sent a curling free kick past the dive of Spurs keeper Guglielmo Vicario who should have done better.
Vicario did much better just before halftime, pushing Cody Gakpo's low shot against the post.
Tottenham should have gone in level at the interval with Richarlison heading a good chance wide and the Brazilian was Tottenham's biggest threat after the break as the visitors grew in confidence and Liverpool slumbered.
Mohamed Salah dragged an effort wide that could have sealed it for Liverpool but the hosts looked uneasy and when Randal Kolo Muani was allowed to dribble across the area he rolled a pass to Richarlison who made no mistake.
It was the 15th goal Liverpool have conceded later than the 75th minute in the Premier League this season, the joint most.
"Being up for so many minutes, yeah of course, two points dropped," Liverpool manager Arne Slot said. "As this season is going, it is not a complete surprise we are conceding in the last moments."
Both sides face big nights in the Champions League on Thursday morning, with Tottenham trying to overturn a 5-2 deficit against Atletico Madrid and Liverpool a 1-0 deficit to Galatasaray.
Tottenham then face a huge home clash against 17th-placed Nottingham Forest next week.
