The Premier League's basement club fell behind to a Fernandinho strike in the second minute of Wednesday's match, before Connor Wickham put Sunderland ahead with two goals in 10 second-half minutes.
Goalkeeper Vito Mannone's spillage from a Samir Nasri effort in the 88th minute cost the visitors victory, leaving Sunderland six points adrift of safety.
But Poyet insisted he was happy to still be standing after a match that burst into life in the latter stages.
"I haven't had the heart attack that football is trying to give me every game," he was quoted as saying by BBC Sport. "We deserved better.
"We conceded a goal after two minutes at Manchester City and every single person in the world expected us to lose from there.
"But we passed the ball and created chances. We kept believing."
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