The south London club were rocked by Tony Pulis' departure by mutual consent on the eve of the season on Friday, but they performed admirably before taking a shock lead at the Emirates Stadium on Saturday.
Having comfortably absorbed waves of Arsenal pressure, debutant Brede Hangeland headed Palace and their caretaker boss Keith Millen into a 35th-minute lead.
Laurent Koscielny levelled on the stroke of half-time, but the FA Cup holders looked like they would fail to capitalise, even as Palace winger Jason Puncheon departed for a second booking in the closing stages.
But Ramsey was on hand to prod home from point-blank range when Mathieu Debuchy's shot was saved, following a display of set-piece defending that would have left Pulis horrified.
Hangeland joined fellow defender Martin Kelly and striker Fraizer Campbell in making his Palace debut, while Arsenal were unchanged from last weekend's Community Shield win over Premier League champions Manchester City.
Arsene Wenger's team tore forward during the early stages in the hope of capitalising on any vulnerability after a turbulent 24 hours for their London rivals.
But the massed ranks of Palace defenders retained the hallmarks of their former boss and Ramsey's tumble under a challenge from Scott Dann for an optimistic eighth-minute penalty shout underscored a frustrating opening for the hosts.
Palace goalkeeper Julian Speroni had to wait until the 29th minute to make his first save - parrying smartly behind after Jack Wilshere's drive bounced awkwardly.
Marouane Chamakh tried to embarrass his former club with an audacious attempt from distance that Koscielny cleared to safety after goalkeeper Wojciech Szczesny tore out of his area to challenge Campbell.
The incident appeared to unsettle Arsenal and they were still reeling when Hangeland shrugged off Alexis Sanchez and leapt above Koscielny at the near post to glance home Palace's first corner of the match.
But the roles were reversed in first-half stoppage time when Koscielny peeled away from Hangeland to nod Sanchez's floated free-kick beyond Speroni.
Arsenal left-back Kieran Gibbs limped off with a suspected hamstring problem to be replaced by Nacho Monreal seven minutes into the second half, by which time the match had settled back into the pattern established the last time it was level.
Sanchez drove at the Palace defence in the 68th minute and cleverly exchanged passes with Santi Cazorla, but Speroni's save was routine.
Olivier Giroud and Alex Oxlade-Chamberlain emerged from the bench as Arsenal pushed for a second goal, the former agonisingly close to turning home a close-range winner in the 75th minute.
Arsenal's efforts appeared in vain until Ramsey's late intervention. Palace showed plenty of the spirit that Pulis fostered in escaping relegation last term and will hope this result is not an indication of how much they will miss his Midas touch.
