The bottom club's hopes of ending a miserable eight-game losing streak evaporated in a crazy end to the first half with Emiliano Buendia sent off after 35 minutes for an elbow and then Josip Drmic following him 10 minutes later for a lunging tackle.
Norwich showed plenty of spirit after the break to frustrate Burnley but their wretched afternoon was summed up in the 80th minute when defender Ben Godfrey turned a cross into his own net.
Sixth spot will earn a Europa League place.
Norwich became the third team in English top-flight history to lose five consecutive home games without scoring a goal and the first since Birmingham City in 1922.
The Canaries also extended their record league losing sequence to nine and will end the season with the lowest points tally in the club's history.
Burnley have set records too but more positive ones as for the first time they have won 15 Premier League games in a season. Goalkeeper Nick Pope also leads the way for Premier League clean sheets this season with 15.
"I don't think I've played against a team with two sendings off. It was a bizarre game. We'll take the clean sheet and the win," Burnley boss Sean Dyche said.
"To have 15 clean sheets and the wins on the table is very pleasing so I'm delighted for the players."
While Norwich have earned respect for their attractive style of play this season, if only 21 points with one game remaining, their defeat at Carrow Road was ugly.
Buendia needlessly brought his elbow down on Ashley Westwood's head and was shown a red card after referee Kevin Friend watched the incident back on a touchline monitor.