The defender, making his first Premier League start of the season, powered home with two minutes remaining at Old Trafford.
Hause then looked like going from hero to villain when he handled in the box to concede a stoppage time penalty.
But Bruno Fernandes blazed the spot-kick over the bar as Villa held on for their first win at Old Trafford since 2009.
That win, earned by the same scoreline with Gabby Agbonlahor Villa’s match-winner, was their only Premier League win over the Red Devils for 26 years.
Little wonder the away section of Old Trafford exploded with delight at the final whistle. This was a real “I was there” occasion.
And Villa deserved their win, which might just be the most impressive of Dean Smith’s three-year reign to date.
Though the stats might show the hosts had comfortably more possession and shots, they do not tell the story of a game which ebbed and flowed and which Villa were always at least United’s equal.
Both Matt Targett and Ollie Watkins missed golden chances to break the deadlock in the first half, while Jacob Ramsey slipped at the vital moment when it looked like he might put Villa ahead late on.
It mattered not as Hause got ahead of Edinson Cavani to nod home the opener in the 88th minute. When the defender then conceded a spot-kick it looked as though Villa would be denied but Cristiano Ronaldo allowed Fernandes to take it and it proved a costly error.
Fernandes wasted a good chance for the hosts inside the opening 90 seconds when he controlled Luke Shaw’s low cross on the edge of the box but under modest pressure from Douglas Luiz fired hurriedly over the bar.
Ronaldo also slapped a shot wide from distance but it while it was United dominating possession, the best chances fell to Villa.
The first followed great work between John McGinn and Matty Cash, the former sending the latter racing into the box with a perfectly-judged pass through the backline. When Cash then sent in a fizzing low cross Targett looked certain to convert but fired his finish way over the bar.
Villa’s next chance was handed them on a plate by the hosts. Harry Maguire, from out wide, drilled a pass back to David De Gea which the keeper could only deflect into the path of Watkins, 15 yards out. The striker had time to set himself but hit the finish straight at the United keeper, who saved with his legs.
In truth, it wasn’t only Villa who were missing their shooting boots. Mason Greenwood also fired weakly at Emi Martinez after a rapid counter from the home side, while the visitors were left breathing a sigh of relief when Pogba’s shot hit the leg of a defender and squirted wide.
Ezri Konsa was the next man to go close for Villa, heading over from Luiz’s corner, with Aaron Wan-Bissaka doing enough to put him off.
Having had the better chances the visitors were perhaps fortunate to get to half-time with the scores level.
Martinez was called into his first serious action of the afternoon when he saved from Maguire after the United captain had been left unmarked at a free-kick, before Pogba sent a header inches wide when Raphael Varane flicked on a corner.
There was a sense half-time came at the right moment for Villa and they were much the better team for the opening 15 minutes of the second period, without carving out a gilt-edged chance. Watkins saw appeals for a penalty waved away after he appeared to be bundled over in the box.
Instead it was United who started to get the chances, Pogba lifting a shot over the bar after being teed up by McTominay, before Greenwood sent a shot too close to Martinez after being found in space by Fernandes.
United, who had earlier lost Luke Shaw to injury, saw his England team-mate Maguire leave the field just past the hour mark after picking up a knock.
A thrilling passage of play then saw chances at both ends. First, De Gea beat away Watkins’ powerful drive after a superb Luiz pass had sent him scampering away. Seconds later Greenwood fired wide as United looked to strike on the counter.
Pogba wasted a good chance for the hosts when he fired wide of the near post having been picked out by another smart Fernandes pass. The Frenchman then shot straight at Martinez after creating his own opening with a jinking run.
Smith introduced Buendia off the bench and he looked like setting up the opener when he rolled the ball into the path of Ramsey but the youngster slipped before it reached him and the chance was gone.
Instead the moment belonged to Hause. With two minutes remaining Villa won a corner and when Luiz whipped it in from the left, the defender got ahead of Edinson Cavani to power a header home.
Even then it looked like Villa would be denied when Hause handled Fernandes’ cross and Mike Dean pointed to the spot. But the United playmaker, usually so dependable, blazed his spot-kick over the bar. Finally, this was Villa’s day.
88 GOAL Kortney Hause heads Villa in front, getting in front of Edinson Cavani at a corner and glancing home.
90+3 PENALTY MISSED Bruno Fernandes is handed the chance to salvage a point for United when Hause handballs but blazes his penalty over the bar.
Man Utd (4-2-3-1): De Gea, Wan-Bissaka, Varane, Maguire © (Lindelof 66), Shaw (Dalot 34), McTominay (Cavani 80), Fred, Greenwood, Fernandes, Pogba, Ronaldo Subs not used: Martial, Lingard, Sancho, Matic, Van De Beek, Heaton (gk).
Villa (3-5-2): Martinez, Konsa, Mings, Hause, Cash, McGinn, Luiz, Ramsey (Archer 85), Targett, Ings (Buendia 78), Watkins Subs not used: Bogarde, Young, Nakamba, Chukwuemeka, Traore, El Ghazi, Steer (gk).