Palace were trailing 3-1 when skipper Michael Hughes pulled one back in the 73rd minute and then, with just seven minutes remaining, Andy Johnson converted his tenth penalty of the season to ensure the spoils were shared.
Perhaps the point is not going to be enough to save Norwich from the drop, but it may just keep alive Palace's chances of survival.
A good but edgy first half was followed by a magnificent second period which kept the near capacity crowd on the edge of their seats, right to the bitter end.
Joonas Kolkka gave the Eagles a great start with a goal after just five minutes.
Nervous Norwich had already conceded a number of free-kicks, and Palace made them pay with Hughes floating his kick over to the far side where the flying Finn struck a superb half-volley from an acute angle, which flew past Robert Green.
Palace were firing on all cylinders and ten minutes later Dougie Freedman looked set to add to Palace's lead only for Green to pull off a remarkable one-handed diving save.
Norwich were gradually getting back into the game, and Thomas Helveg fired just wide after Palace had charged down a free-kick. It was a tense affair with so much at stake, and Johnson was looking for a penalty after being floored on the byline by Jason Shackell, but referee Rob Styles was having none of it.
Norwich survived that scare and regrouped as they began to hit back.
Leon McKenzie was looking particularly lively up front and the ex-Palace man sprinted past two defenders on the left-hand side and crossed for Dean Ashton to tap home his second goal since moving to Carrow Road from Crewe in January.
Norwich turned the screw after the break. Straight from the restart, Damien Francis found Ashton. He got the better of Gonzalo Sorondo in the chase for the ball, before firing home just 11 seconds into the second half.
Eight minutes later the Canaries were in heaven as Darren Huckerby got past Danny Butterfield and neatly crossed into the goalmouth for McKenzie to steer the ball past Gabor Kiraly to open up a 3-1 lead.
The 2,900 visiting fans were having a rare treat as their first away win beckoned.
Palace were plugging away, but Norwich were snapping at their heels, and space was limited.
With 17 minutes left, Palace finally put a string of passes together and Sandor Torghelle found Johnson on the right-hand side. The Palace hit-man struck his cross to Hughes who stooped to power home a header that gave Green no chance.
Game on, and nine minutes later Palace were level. Johnson at full speed collected a hopeful pass from Tom Soares but his advance on goal was terminated as Shackell made an unnecessary challenge to concede a penalty.
Johnson, somewhat shaken by the challenge, made no mistake from the spot to score his 20th goal of the campaign.
The Eagles pressed for a late winner and Soares' overhead kick was stopped by Green. The keeper was in action again with a brilliant diving save to deny Freedman in stoppage time.
MAN OF THE MATCH: Michael Hughes (Crystal Palace) – Just when it looked as though the Canaries were going to leave Selhurst Park with three vital points, the Northern Ireland international stooped to head home one of the most important goals of his career.