Midfielder Ambrose rifled in his fourth goal of the season midway through the second half and Easter settled matters from the penalty spot.
Adkins can seemingly do no wrong at the moment with his team sitting pretty at the top of the npower Championship. His side found themselves in the fourth round and facing a winnable tie against third-placed Palace at Selhurst Park, but Adkins made 10 changes to his line-up - despite an impressive debut from 16-year-old James Ward-Prowse - and came up just short.
Instead it was a highly-rated Palace youngster, Jonathan Williams, one of seven changes for the hosts, who proved the difference with a hand in both goals. It could have been very different, though, had Saints midfielder Jack Cork's header not been cleared off the line by Stuart O'Keefe in the opening minute.
Palace's back-up goalkeeper Lewis Price also had to get down sharply to keep out a snap-shot from Ben Reeves and block a fierce drive from Jonathan Forte.
Palace were restricted to long-range efforts in the first half, none of which came close to troubling Bartosz Bialkowski in the Saints goal. Nathaniel Clyne ballooned a shot five yards over, Easter was even further wide and Kagisho Dikgacoi's drive found the upper tier behind the goal.
Easter should have broken the deadlock 15 minutes into the second half when Williams and Ambrose combined to tee him up just inside the box, but this time the striker dragged his shot wide. It seemed either a moment of magic or a mistake was going to spare the teams, and fans, from extra-time.
And it was the latter which finally did the trick in the 72nd minute as a sliced clearance from Aaron Martin wrong-footed the Saints defence and allowed Williams to get in round the back.
The teenage midfielder sent in a low cross was perfectly weighted for the onrushing Ambrose to tuck past Bialkowski.
Nine minutes later Dan Harding brought down Williams inside the area and Easter stepped up to send Palace through to the last eight for the first time in eight years.