Eden Hazard provided the 17th-minute cross from which Frank Lampard cancelled out Jozy Altidore's opener, and the Belgian then produced two sublime pieces of skill to get his own name on to the scoresheet twice, after 37 and 62 minutes.
John O'Shea had levelled in between Hazard's efforts, and it was fellow Black Cats defender Phil Bardsley who had the final say at both ends with an 84th-minute own goal and then Sunderland's third seconds later.
Poyet, whose side remain bottom of the table, said: "If we can play like this and fight like this and make it so difficult like we did here, we will stay in the Premier League for sure.
"Maybe we needed one more corner, one extra corner in the last six minutes. But we didn't get it."
The Uruguayan was as impressed as Mourinho with Hazard's contribution, admitting: "He was unplayable, Hazard.
"I don't remember a player performing at this level against me as a manager. He was spectacular.
"We tried everything - doubled up, tripled up, we changed the player who was playing wide. We nearly put on another full-back, but we were losing, so we couldn't.
"It was a great display, probably the best I have seen this season from an individual player."