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The result sees United fall out of the top four and four points behind leaders Liverpool.
Leicester went close to opening the scoring after five minutes but Harry Maguire cleared off the line to prevent Vardy from scoring into an empty net.
The hosts dominated the early stages but United took the lead after 19 minutes when Greenwood exchanged passes with Bruno Fernandes and fired a stunning strike into the top corner.
The game came to life with 12 minutes remaining when Soyuncu fired home from close range from a corner to put Leicester in the lead.
Four minutes later, substitute Rashford made it 2-2 when he latched onto Victor Lindelof‘s pass.
However, Leicester went back ahead seconds later after Vardy pounced onto Ayoze Perez‘s ball.
Daka sealed the win for Leicester in stoppage time with a header at the back post to increase pressure on Solskjaer.