Ibrahimovic netted the winner in the 88th-minute from a tight angle after Paul Pogba slickly threaded the ball through to the 35-year-old striker.
"I told him England is not the best place for a holiday for a rich guy," United manager Jose Mourinho said of the Swede, who joined from Paris Saint-Germain in July.
"You come to England for the most difficult league and to prove yourself. It was a decision that was based on that. It's not just about his goals. It's his leadership, motivation."
Thanks to Ibrahimovic, Mourinho has consecutive wins in the league for the first time since August to keep United three points behind fifth-place Tottenham.
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Palace is only three points above the relegation zone after one win in 10 matches, perilous form for manager Alan Pardew who has spoken of his job insecurity in recent weeks. Palace was gifting United space and possession for much of the first half, rarely breaking out of its own half.
United was fortunate to remain with 11 men after Marcos Rojo's two-footed challenge on former Old Trafford winger Wilfried Zaha. The defender escaped with a yellow card.
"He's an aggressive, clean player," Mourinho said, "probably playing the best football of his career."
It's not the best from Palace this season.
In the second minute of stoppage time at the end of the first half, Palace was punished by its inability to defend a set-piece but the hosts had legitimate grievances on two counts.
The home fans were furious, jeering referee Craig Pawson at the interval. The players drew on the anger.
Joe Ledley, a halftime replacement for Mathieu Flamini, immediately came close to leveling. It was a sign of the threat, missing for so much of the first half from Palace. The first significant contribution from United goalkeeper David de Gea was a save at full stretch to turn McAthur's shot over. The leveler was stylish. Damien Delaney received the ball from Joel Ward, and a back-heeled flick-on enabled McArthur to slot past De Gea in the 66th minute.
And if United was fortunate to benefit from refereeing decisions in the first half, the fortunes were now reversed. From the corner after Hennessey's save, Ledley raised an arm to block the ball. Pawson was surrounded by United players but no penalty was awarded.
Mata did soon have the ball in the net but he was harshly adjudged to be offside after being set up by a downward header from Rojo.
Ibrahimovic's late intervention ensured United left south London relieved rather than furious.
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