Joshua King rifled the Cherries ahead after nine minutes before Artur Boruc was then forced to save Yohan Cayabe’s spot kick just minutes later, with Charlie Daniels adjudged to have held his man inside the area.
But it just wasn't to be for the Cherries, as a late bout of pressure eventually paid off and Dann rescued a point for the hosts.
Eddie Howe opted to keep faith in the side that narrowly lost out at West Ham United last weekend, with Harry Arter named in the starting XI despite his sending off, after he served his one-match suspension in the midweek EFL Cup tie at Morecambe.
And Howe’s decision appeared to pay off early on as the Cherries started the game on the front foot, with both Daniels and Ryan Fraser getting crosses in early on – only to see them cleared.
The goal did come on nine minutes though, as Harry Arter picked up a cleared free kick and played in King, who slotted the ball into the far bottom corner.
The Cherries were good value for the lead – but it was almost taken away from them six minutes later as referee Dean penalised Daniels for holding on to his man inside the area.
But Boruc, who has saved his past three penalties in a Cherries shirt, once again went the right way and got down to his left to palm Cayabe’s penalty away.
Howe’s men were well on top as the frustration of the Palace fans became evident, and Andrew Surman was the next player to come close as he sent a free kick narrowly over the bar.
Palace did have their own glimpses of goal and Boruc again was the hero shortly after as, despite being off balance, he tipped over a teasing effort from summer signing Andros Townsend.
Callum Wilson almost doubled the Cherries’ lead on 35 minutes as he combined well with King, but the striker couldn’t poke the ball beyond the on-rushing Steve Mandanda when through one-on-one.
There was cause for concern for the Cherries shortly before the break, as captain Simon Francis appeared to land awkwardly under a challenge inside the Palace half. Fortunately, despite holding his shoulder in the aftermath, he was ok to continue.
Palace appeared rejuvenated for the second period and the Cherries had to withstand long periods of pressure, as they flooded forward.
Their most clear-cut chance came from Scott Dann, as his goalbound header floated towards the line, only for Arter to clear it just before it crossed the line.
In contrast to the first half, the Cherries were forced to play largely on the break as Palace looked to find a way back into the game and on 56 minutes Howe noted the danger and brought on Dan Gosling in place of Ibe.
Steve Cook, who had an impressive afternoon, was the next player to save his side, as he hooked a Wilfred Zaha header across the box behind for a corner.
Zaha, introduced by Alan Pardew in the second period, continued to look lively and on 73 minutes he was left frustrated as fellow substitute Lee blasted over, after he had pulled the ball back to him inside the area.
Lewis Grabban was also introduced by Howe in the second period as the Cherries reverted to a 4-5-1 formation, and he had their best chance of the second period as he picked up on a ball over the top and attempted to poke it over Mandana, only for it to land in his arms.
Ryan Fraser also came close as he met Adam Smith's drive into the area, but could only tap the ball just wide at the near post.
The hosts showed no signs of letting up and relentlessly attacked the Cherries goal, only to be denied by blocks from Steve Cook and Boruc.
But they did eventually get the breakthrough, though. A deep cross into the area was met by Dann who powered a fine header into the far corner in the final moments to secure the hosts a share of the points.
Crystal Palace: Mandanda, Ward, Dann, Delaney, Souare, McArthur (Lee 59), Puncheon, Cabaye, Townsend, Wickham (Zaha 59), Benteke.
Substitues: Speroni, Tomkins, Kelly, Ledley, Boateng.
Cherries: Boruc, A. Smith, Francis, Cook, Daniels, Surman, Arter, Fraser, Ibe (Gosling 56), King, Wilson (Grabban 69).
Substitutes: Federici, B. Smith, Ake, Gradel, Afobe.
Attendance: 23,503 (1,966 away).
Referee: Mike Dean