Palace, under the caretaker guidance of Keith Millen following Tony Pulis’ shock departure, took the lead in an opening weekend derby date in north London through Brede Hangeland, but were pegged back before the interval by Laurent Koscielny.
Millen’s side were then reduced to 10 men in the closing stages as Jason Puncheon was shown a second yellow card, and Arsenal made their numerical advantage count almost immediately as they pounced in stoppage-time.
The opening two goals came from set-pieces, with defensive errors cancelling each other out.
On 35 minutes, from a Puncheon corner, Hangeland was allowed to run away from Alexis Sanchez and Koscielny and glance past Wojciech Szczesny at the near post.
Then, on the stroke of half-time, Hangeland returned the favour as he failed to pick up a run from Koscielny and was left to watch on helplessly as the Frenchman nodded a Sanchez free-kick into the bottom corner.
Palace dug in from that point, but suffered a late blow as Puncheon slid in to catch Nacho Monreal on the ankle and was given his marching orders.
Arsenal, who had been far from their free-flowing best, seized their opportunity in the dying seconds as Ramsey bundled home from close-range after an Alex Oxlade-Chamberlain cross had caused confusion inside the box and Mathieu Debuchy had seen an effort parried by Julian Speroni.
Arsenal made a positive start and Sanchez, signed from Barcelona over the summer, tried his luck when the ball broke to him some 25 yards out, but the Chilean's long-range chip over Speroni drifted well wide.
The Palace goalkeeper then had to be alert on 29 minutes to get down quickly and push a low, curling 18-yard shot from Jack Wilshere behind.
There was a rare moment of panic in the Arsenal rearguard when Szczesny's scuffed clearance was driven back towards goal from the halfway line by former Arsenal forward Marouane Chamakh.
Koscielny was able to get across and block, before Fraizer Campbell's follow-up was put behind.
From the corner, Palace took the lead on 35 minutes. Puncheon floated the corner in from the right, where Hangeland flicked it on at the near post into the far corner.
After being hit against the run of play, Arsenal needed to regroup quickly, and did so as Koscielny levelled just before half-time.
Sanchez was fouled some 35 yards out, and floated a deep free-kick over the Palace defence.
Koscielny ghosted behind the wall of yellow shirts and flicked the ball off the back of his neck into the corner.
Arsenal were forced into a change just after the start of the second half when full-back Kieran Gibbs pulled up with what looked like a hamstring problem and was replaced by Monreal.
Palace continued to prove difficult to break down, which had been a trademark of Pulis' reign, but also looked dangerous on the counter.
On the hour mark, Yaya Sanogo was replaced by Olivier Giroud, the France international having come off the bench at Wembley last weekend to crack home a superb third goal in the Community Shield win over Manchester City.
Arsenal appealed for a penalty when the ball bounced onto Chamakh's arm and chest, but the referee was unmoved.
Chamakh, who had committed a string of fouls all afternoon, was then finally shown a yellow card for clattering into Wilshere on the halfway line.
The midfielder was soon replaced, perhaps as a precaution, by England team-mate Oxlade-Chamberlain with 20 minutes left.
Arsenal had another penalty claim turned away by referee Jonathan Moss when Giroud looked to have been pulled down by Hangeland as the ball was swung in from Sanchez.
Giroud flashed an angled drive across the face of goal before, with two minutes left, Puncheon was sent off for a second caution after a late challenge on Monreal.
Arsenal continued to press and were rewarded in the closing moments.
Giroud's knockdown was fired goalwards by Debuchy and although Speroni produced a fine reaction stop at point-blank range, Ramsey was alert to smash in the rebound and send the Emirates Stadium faithful into wild celebration.