Alexis Sanchez gave Arsene Wenger's side the lead in first-half stoppage-time when he ghosted behind the Eagles' defence to lift a header over Wayne Hennessey.
However, Palace were resolute defensively and thwarted Arsenal on several occasions, and finally took their chance when it came their way when Bolasie crept a short under Petr Cech.
Changes to line-up fail to make a difference for Palace
Palace made two changes to the side that drew against Everton in midweek with Joe Ledley and Connor Wickham replacing Wilfried Zaha and Dwight Gayle - who both picked up knocks in the 0-0 stalemate. Zaha dropped to the bench while Gayle missed out on a place in the squad completely.
The early stages of the game were dull and neither side were able to break through the respective back fours. Alan Pardew's decision to opt a 4-4-2 appeared as though it could be shrewd as the Gunners failed to break down the Eagles' two banks of four, but whenever Palace won possession, Bolasie and Wickham failed to take advantage further up the pitch.
Arsenal's quality and the Eagles' nervousness were equally evident as the first-half wore on. Palace struggled to keep the ball throughout the opening period - especially due to the lack of movement from Bolasie and Wickham - and Arsenal capitalised by creating several half-chances, particularly for Danny Welbeck and Sanchez.
Mesut Ozil had his side's first clear-cut chance with just two minutes remaining in the first-half when Alex Iwobi's pass put him through on goal, but Hennessey did well to thwart the midfielder, although the effort was poor.
Palace finally responded and had their best spell of the game in their latter stages of the half. Wickham had a glorious chance to head Palace into the lead but he miss-timed his jump and the header dipped over the untroubled Cech's goal.
Poor defending costs Palace once more
Despite the chance, Palace had ridden their luck and were made to pay just a couple of minutes later when Mile Jedinak dawdled on the ball on the edge of his own penalty area, and he was punished for his sloppy play when he was dispossessed by Welbeck, who crossed for Sanchez to head over Hennessey and into the back of the net.
Jedinak was replaced by Bakary Sako at half-time following a poor first-half performance as Palace aimed to be more adventurous in Arsenal's half of the pitch in the second period of the game.
However, the second-half started in a similar fashion to how the first-hald ended with Arsenal the dominant side. Sanchez went close to a second goal and Welbeck should have done better with an effort that Hennessey caught easily at his front post.
Ozil almost troubled Hennessey once again with a 30-yard free-kick, but he was unlucky to see his effort fly just wide of the Welsh goalkeeper's goal.
Adebayor introduction pays dividends against former club
Emmanuel Adebayor's introduction was an attempt to increase the tension in the atmosphere inside the Emirates, but the former Arsenal man almost gifted his old teammates a second goal when his sloppy clearance deflected off Welbeck and Hennessey had to rush out and deny the Arsenal forward from scoring.
Adebayor's first involvement in the attacking third resulted in a shock equaliser as he broke away with the ball following an Arsenal attack. However, he was forced to cut back inside and found Bolasie who cut across the Arsenal defence before firing a shot into the bottom corner and past Cech - who will feel he should have saved the effort.
The goal, albeit against the run of play, gave Palace increased confidence and the introduction of Zaha almost won the game as the winger outpaced Nacho Monreal before getting past Laurent Koscielny, but could not find Adebayor across goal who would have had a simple tap in.
Palace will be delighted with their point, despite it being slightly undeserved, and move onto 39 points which should equate to safety.