Palace required a single point to make certain of Championship survival despite having 10 points deducted.
Alan Lee gave them an early lead but Leon Clarke levelled, before Darren Ambrose restored the lead after 62 minutes.
Darren Purse levelled again four minutes from the end but it was not enough to save the Yorkshire side.
From the start Palace eased into control and looked composed. Darren Ambrose smacked a free-kick into the Wednesday wall and then forced a great save from Lee Grant with another super effort.
And after 24 minutes the visitors were in front as this time Ambrose swung over a perfect corner and striker Lee rose above the rest to power in an accurate header.
The Yorkshire side bounced back and Johnson smashed another shot that deflected wide before Mark Beevers headed straight at the keeper.
Wednesday looked for a penalty when Leon Clarke went down in the area under a Patrick McCarthy challenge.
Two minutes before half-time, Danny Butterfield was caught out under pressure from Clarke, who rounded him and clipped an accurate shot that brushed Julian Speroni's outstretched fingers but curled in.
It was Clarke's last contribution before he was stretchered of moments later with an ankle injury.
Tempers flared as the players left the pitch at half-time after a clash between Palace midfielder Shaun Derry and Wednesday's James O'Connor.
O'Connor returned after the break with his head bandaged and Derry was in the thick of it in the second half in a clash with Darren Potter, who had to take a break for stitches in a head wound.
Fourteen minutes into the second half, Wednesday came close to taking the lead when Luke Varney crashed a shot that was blocked.
But in the 62nd minute Sean Scannell drove a low cross and as Wednesday dithered Ambrose struck with an accurate shot .
In a grandstand finish, Beevers headed on at the near post and Purse pounced to slot in the equaliser.