Back-to-back home draws increased the pressure heading to embattled Sunderland, where United extended their unbeaten league run to 21 matches and bolstered their hopes of Champions League qualification.
Ibrahimovic opened the scoring with a ferocious strike and, having seen Sunderland's Seb Larsson sent off and Henrikh Mkhitaryan add a second, he provided the assist for Marcus Rashford to wrap up victory.
It took the 35-year-old's goals tally to 28 for the campaign and led the brash striker to liken himself to Brad Pitt's title character from the film The Curious Case of Benjamin Button, who ages in reverse.
“I train hard, I keep focusing, I believe in what I do,” Ibrahimovic told Sky Sports. “It seems like the older I get, the more intelligent I get. You don't waste so much energy. I feel like Benjamin Button - I was born old and I will die young.”
Mourinho praised Ibrahimovic after a win that reduced the gap to fourth-placed Manchester City to four points, with United also boasting a game in hand.
“After Liverpool and Manchester City's victories, if we don't win today it's almost mathematically impossible (to finish fourth),” he said. “It was important for us to win and that was the main objective.
“We did that and, apart from that, no more injuries to add to a difficult list we have because we don't hide that we are still fighting for our situation in the Premier League.
“But Europa League becomes fundamental for us, it's the only competition we can win now,” he said. “We did the Community Shield, we did the League Cup. Premier League, no way, so it is our competition to fight for.”
The win on Wearside lifted the mood and even saw Mourinho praise Luke Shaw, having responded impressively to recent criticism.
“He played well, played solid,” Mourinho said. “The opposition was not creating big problems, but he was solid and was confident with the ball.
“He was going forward with some danger, he was reading the game well to go when Mkhitaryan went inside. He played really well, so it was a good hour for him.”
By contrast it was another bleak afternoon for former United boss David Moyes, whose side remain 10 points adrift of safety with just seven games remaining.