Ramsey, whose goal won the FA Cup at Wembley in May, smashed the ball home from close range as Arsene Wenger's side came from behind to win their Barclays Premier League opener 2-1 at the Emirates Stadium.
The defeat was harsh on Palace, who had produced a determined display at the end of a difficult few days following the shock departure of manager Tony Pulis on the eve of the new season.
Indeed, a first-half header from Brede Hangeland, one of the few summer signings sanctioned by chairman Steve Parish, had put the hard-working Eagles ahead before Laurent Koscielny flicked home a free-kick from £30m Chile forward Alexis Sanchez to level before the break.
Arsenal lacked a cutting edge in the final third, creating only a few half-chances before Ramsey was on hand to slot home after Mathieu Debuchy's shot was saved - following Jason Puncheon's dismissal for a second yellow card.
Mauricio Pochettino's first Premier League game in charge of Tottenham Hotspur proved more memorable as defender Eric Dier grabbed an injury time winner in an incident-packed London derby at West Ham United.
The visitors were reduced to 10 men in the 29th minute when Kyle Naughton was sent off for handling a Kevin Nolan shot in the box, but Mark Noble trundled the resulting penalty wide.
The Hammers were also reduced to 10 men in the 63rd minute when James Collins was issued his second yellow card, and things got worse for the hosts as Harry Kane sent Dier clear to snatch the winner on his debut in added time.
Alan Irvine also looked set for a debut win in charge of West Brom before a late strike from Sebastian Larsson salvaged a point for Sunderland in a feisty 2-2 draw.
Lee Cattermole's thunderous early opener was cancelled out by a Saido Berahino penalty and the hosts' striker put his side in front with a close-range volley in the 74th minute.
But Gus Poyet's men refused to accept defeat and grabbed a point when the impressive Patrick van Aanholt crossed from the left and set up Larsson to stroke home the equaliser.
Substitute Chris Wood gave Leicester City something to celebrate on their return to the top flight as his 86th-minute effort sealed a 2-2 draw with Everton.
Aiden McGeady curled the visitors in front on 20 minutes only for Leonardo Ulloa, City's recent capture from Brighton, to equalise almost immediately.
Steven Naismith put Everton back in front on the stroke of half-time but Wood was on hand to hit the leveller past Tim Howard after impressive work from Riyad Mahrez.
Hull City keeper Allan McGregor saved a late Charlie Austin penalty as the Tigers hung on for a 1-0 win at QPR thanks to James Chester's diving header in the 52nd minute.
Andreas Weimann hinted at brighter times ahead for Aston Villa as his 50th minute drive sealed a 1-0 win over Stoke City at the Britannia Stadium.