Andros Townsend's deflected strike in the 11th minute gave Roy Hodgson's side an unlikely lead at half-time before Patrick van Aanholt doubled the scoreline just three minutes after the break.
Chris Smalling and Romelu Lukaku netted to draw the visitors level at with 14 minutes left and then a stoppage time Goal of the Season contender from Nemanja Matić dramatically secured three points.
It takes two to tango
The United manager was unsurprisingly relieved after full-time, but made sure to stress his disappointment in his team's defending.
Speaking to Sky Sports at Selhurst Park, Mourinho said: "I think Palace deserve congratulations, because a match like this is only possible with two teams. It was a fantastic match."
He did, however, rue his side's errors and described the match as "difficult" for him to watch because the Red Devils "made so many mistakes."
United got their act together and reacted well to going two goals behind in the second period; their manager said the football "was very good quality" but couldn't leave the topic of defensive mistakes.
"I don't like my teams to lose their organisation like we did, but it was an amazing match and a very important win for us."
Palace's second "disgraceful and childish"
The United defence, most notably captain Antonio Valencia and match-winner Matić, were caught napping for Palace's second after the latter comitted a foul.
Jeffrey Schlupp quickly slotted the subsequent free-kick through for van Aanholt to latch onto and fire past David de Gea - a moment Mourinho has said was "disgraceful" and "childish."
He added: "The players kept an amazing attitude though. Juan Mata was finding spaces inside, Marcus Rashford was open on the left and brought their right-back wide to give us more space in the middle. You always need that little bit of luck to win a match in the last few minutes."
Despite trailing 2-0, Mourinho insisted he always believed the Red Devils could come back into it and Smalling's 55th minute header was the turning point.
"Their second goal made things much more difficult for us, but Chris Smalling's goal was very important. I always believe we can win. I will be like that until my last day.
"Some days more than others but after Smalling's goal, I always thought it was possible."