A tight first half passed by with just a handful of chances, as neither side were willing to give too much up to their fellow relegation battlers.
Norwich were first to strike in this battle at the bottom, after good work from Wes Hoolahan and Gary Hooper teed up Bradley Johnson, with the former Leeds man finishing with aplomb from 18 yards.
They held the lead for just five minutes, however, after Leroy Fer was adjudged to have pushed Mile Jedinak in the area from a corner, giving Jason Puncheon the chance to dispatch the penalty and score his first Palace goal.
That goal proved to be the last of the game, as Fer was sent off for a second booking late on with both sides chasing what would have been a crucial three points.
Palace started perhaps the brighter in front of their home fans, but the finishing touch that has been missing for them so far this season deserted them once again early on.
Their first real chance came after nine minutes, when Joel Ward crossed to the back post to find Puncheon, but the former Southampton winger fluffed his lines when well placed.
Four minutes later, Yannick Bolasie's power earned a free-kick against Michael Turner, but Jedinak couldn't find enough dip on his effort from 20 yards.
Palace's pressing game continued to put Norwich under pressure, and it showed after 32 minutes when Puncheon robbed Martin Olsson of the ball on the right hand side, but Cameron Jerome couldn't convert the cross when unmarked.
Leroy Fer then hit a tame effort on goal a matter of minutes later as Norwich threatened to break the deadlock after a scrappy opening 40 minutes, a threat which they delivered on just a couple of minutes later.
Danny Gabbidon was the culprit for Palace, as his dithering on the ball allowed Hoolahan to snatch possession, before finding Hooper who laid off to Johnson, with no mistake made on the edge of the area by the midfielder, as the finish curled in off the post.
Palace's frustration at going behind was palpable, and nearly cost them further after Marouane Chamakh grabbed Hoolahan by the throat after a tussle between the two, costing them both a yellow card. Chamakh, however, was lucky to escape without a red.
The hosts applied the pressure just before half time in search of an equaliser and it was Fer who gave them the opportunity on a plate after pushing Jedinak over in the area from Puncheon's corner.
Puncheon stepped up to take the penalty and rifled it home for his first Palace goal.
Palace should perhaps have been ahead before half time, when Jerome nearly headed home Ward's corner, but Olsson was on hand to rescue Norwich and clear off the line.
Tony Pulis' side came bursting out of the blocks in the second half and on 53 minutes they should have been ahead when Jedinak laid the ball off to Jerome just eight yards out, but the striker lashed his finish high and wide.
Palace continued to throw everything at the visitors in the hope of grabbing a victory that would put them above Chris Hughton's side in the table but, despite a succession of corners in a row, the chance would not fall.
Norwich nearly hit them on the break on 69 minutes, too, when Hooper was fed 18 yards out, but Julian Speroni got down well to save to his right.
The hosts were well on top again as the second half went on, and had cries for a handball in the area by goalscorer Johnson from Ward's cross turned down by Mike Dean after 75 minutes.
With both sides chasing the win, frustrations boiled over late in the game when Fer, who had a disappointing game, was dismissed for a two-footed lunge on Jedinak (82).