Substitute Duncan Watmore then added the third goal in stoppage time after poor defending.
Norwich remained on 31 points, one spot above the Premier League relegation zone, while Sunderland moved on to 30.
Norwich have four games left to play, Sunderland have five and now a far better chance of scrambling to safety in the last few weeks of the campaign for the fourth successive season.
Sunderland manager Sam Allardyce, who took over from Dick Advocaat in October when Sunderland were one place off the bottom, said he was delighted his side had closed the gap to one point, while still having a game in hand.
"That was a big, big, big win. That all-important clean sheet was the key element for our victory today," he said.
"We defended very well against their long balls into the box and we got some quality goals and it could have been more in the second half.
"It was a fantastic 3-0 win for us, without us playing all that well, and hopefully spur us on to more victories because we need them to be safe."
The closest the home side came to scoring was when substitute Nathan Redmond hit the post from 30 metres three minutes after halftime and although they had a couple of penalty appeals turned down, Norwich rarely looked like taking anything from the match.
"If we are going to stay up we are going to have to do it the hard way," said a deflated Norwich manager Alex Neill, whose team faces Arsenal, Manchester United, Watford and Everton in the last four games.
(Reporting by Mike Collett; Editing by Tony Jimenez/Alan Baldwin)
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