A deliberate handball by Ashley Williams, who was sent off, allowed Ibrahimovic to cancel out Phil Jagielka's opener yesterday and extend United's unbeaten run in the Premier League to 20 matches.
But it remained an unwelcome result for Jose Mourinho's side, who have now drawn nine games at home this season and trail fourth-place Manchester City by four points.
"The performance from a football point of view was not good, but I am very pleased with the effort," said Mourinho, whose team had drawn 0-0 at home to West Bromwich Albion on Saturday.
"We don't kill off enough opponents and at the end of this season I think we will be punished by that."
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Elsewhere, Leicester City's resurgence under Craig Shakespeare continued as they beat bottom club Sunderland 2-0 to record a sixth successive win since the dismissal of manager Claudio Ranieri.
Ibrahimovic's last-gasp spot-kick -- his 27th goal of the season -- meant United held on to fifth place, three points above Arsenal and Everton having played a game more than the former.
But the Europa League looks increasingly like their best hope of securing a return to the Champions League next season.
United manager Mourinho welcomed top scorer Ibrahimovic and Ander Herrera back from suspension, while Paul Pogba returned to the bench following a hamstring injury.
His side fell behind in the 22nd minute when Williams flicked on Kevin Mirallas's right-wing corner and centre-back Jagielka held off Marcos Rojo to poke a volley between David de Gea's legs.
- Moyes misery -
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Everton goalkeeper Joel Robles, much criticised after his side's 3-1 derby loss at Liverpool, produced flying saves to repel a Daley Blind free-kick and a rasping Herrera drive.
"With the VAR (video assistant referee), we win this game 2-1 because it's not an offside," Mourinho told BT Sport.
But in the fourth minute of injury time, Williams was penalised for parrying a shot from substitute Luke Shaw -- brought in from the cold by Mourinho -- and Ibrahimovic dispatched the penalty.
"It was really disappointing not to get the win," Everton manager Ronald Koeman told BBC Sport.
"The penalty was the right decision, but it was really hard to take."
Substitute Islam Slimani opened the scoring with a 69th-minute header before the revitalised Jamie Vardy smashed in his sixth goal in seven games from Marc Albrighton's pass.
Defeat completed a miserable few days for Sunderland manager David Moyes, who had to apologise after being caught on camera threatening to "slap" a female television reporter.
"I'm just grateful for the club's support," Moyes said. "As far as I'm concerned the matter's finished."
Sunderland lie eight points from safety with eight games of the season remaining.