Leadbitter tucked away his first goal since January from the penalty spot after Claude Davis was sent off, and Edwards finished from close range as Roy Keane's Ipswich maintained their solid start to the Championship season.
The visitors also ended the game a man down after Tottenham loanee Andros Townsend saw red for a foul on goalkeeper Julian Speroni, before Neil Danns pulled one back in stoppage time.
Davids, the 37-year-old Dutch legend who counts Inter and AC Milan, Juventus, Tottenham and Ajax among his former clubs, was in the crowd at Selhurst Park after coming out of retirement to seal his surprise move to south London.
'Edgar the Eagle' was introduced to the Palace fans at half-time, and his debut will be all the more eagerly anticipated after the hosts fell apart in the second half.
There was a definite feelgood factor around the ground after the club finally came out of administration this week.
But it was Town who made the livelier start with Jon Stead forcing an early save from Julian Speroni before Edwards skied a shot.
Townsend drilled a long-range effort wide and Speroni was back in action when he tipped a looping cross from Stead against his own crossbar.
Palace youngster Kieron Cadogan fired high over the bar as the hosts briefly threatened, but the game turned three minutes after the interval when Stead burst into the penalty area.
Davis pulled the Ipswich striker back and was shown a straight red card by referee Jon Moss, while Leadbitter stepped up to coolly send Speroni the wrong way from the spot.
Palace boss George Burley was forced into a reshuffle with defender Adam Barrett replacing Cadogan.
But before they could settle Ipswich forced a second goal, Leadbitter swinging over an inviting cross which Edwards toe-poked into the net at the far post.
The visitors could have had a third goal in the 64th minute when Edwards crossed and Gareth McAuley put his header wide.
A flowing Palace move then ended with Andy Dorman shooting narrowly over, before referee Moss evened up the numbers.
The official had little choice but to send Townsend off after the Spurs youngster's high foot caught Speroni in the head, leaving the Palace goalkeeper in a heap.
The Eagles grabbed a deserved consolation but Danns' injury-time header from Nathaniel Clyne's cross came too late to spark a fightback.