Hause turned from hero to villain in the last three minutes of extra time when he was deemed to have blocked replacement Edinson Cavani’s header with his arm.
VAR upheld the disputed on-field decision, but Villa thought justice was done as penalty specialist Fernandes sailed horribly over the goal to destroy United’s chances of rescuing a point after keeping the ball from Cristiano Ronaldo.
United’s first league setback was made worse by injuries to Luke Shaw and Harry Maguire, dropping Solskjaer to fourth in the table after three defeats in four games in all competitions.
Villa has moved up to eighth place after winning back-to-back games for the first time this season.
Within the first two minutes, Solskjaer expected a quick start from his players, and Fernandes almost delivered when he volleyed over from Shaw’s cutback.
A last-ditch block by John McGinn stopped a long-range attack.
In a dominating first quarter of an hour from Manchester United, Mason Greenwood came within a whisker of striking the target, while Emiliano Martinez watched Ronaldo’s initial attempt slip wide.
As United attempted to respond, Greenwood put Martinez to the test, but Villa continued to generate better chances.
De Gea had little choice but to direct Maguire’s hefty pass back to Ollie Watkins, who had Villa’s second chance, but the Spaniard made amends by making a key stop with his legs.
When Shaw was taken out with an injury 11 minutes before halftime, United’s task became much more difficult.
Diogo Dalot replaced him, but the hosts’ defensive problems persisted, with Ezri Konsa shooting a free header over the bar from close range on 39 minutes.
As United launched a late push on victory, Solskjaer substituted Cavani for the last eight minutes, with Fernandes’ near-post header forcing Martinez into action.
Villa broke the deadlock when Hause dodged Cavani to force a header past De Gea, before Fernandes’ penalty miss secured the win for Dean Smith’s team.