Robert Snodgrass opened the scoring for the hosts on 20 minutes before Alex Tettey scored an incredible volley from 30 yards shortly after the half-hour mark to ensure a comfortable 2-0 win for Norwich.
The result leaves 18th-placed Sunderland three points adrift of safety following a run of just one point from their last four top-flight outings, but Poyet can take solace in the fact his side have two games in hand on the three teams immediately above them.
Sunderland's run-in includes matches against the top three of Chelsea, Liverpool and Manchester City - as well as Everton, Tottenham and Manchester United - but Poyet is still confident of avoiding the drop with 10 games left.
He said: "When we only had one point earlier in the season when everyone else had seven or eight, I was worried. But now we are back in the mix with games in hand.
"Of course, I try to do my best but I'm not that worried. The situation is that if we keep looking at the rest and thinking other results were good for us today, then we are going to forget that we need to play ourselves, for 90 minutes every game.
"It doesn't matter who you play against or where you play, you need to make sure you do your bit because that's what you get paid for."
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