The 22-year-old has signed a four-year contract with Palace and is looking forward to working with boss Ian Holloway.
He told the club's official website: "I'm ecstatic really. I just want to get pre-season started now.
"The chance came and I had to grab it because I might not get another chance. To be playing in the Premier League from where I've come from is a big achievement really.
"I'm just really looking forward to it and hopefully I can take the opportunity with both hands and do well.
"Ian Holloway is good manager and for him to point out my talent and my ability is a good feeling. I just hope I can repay him with goals for the club and help them stay in the Premier League.
"He's built a very attacking side, which is what you need really. You need to score goals to win games to keep you up and to win trophies, which he proved last season."
Gayle originally joined Peterborough on loan from Dagenham and Redbridge in November 2012 before making the move permanent for a fee of £500,000 during the January transfer window.
He made a big impact at London Road with 13 goals in 29 appearances but was not able to prevent their relegation to League One.
The transfer to Palace could reportedly be worth up to £8.5m to Peterborough when add-ons are included, while the Daggers stand to net a windfall after inserting a 20 per cent sell-on clause when Gayle moved to Posh.
Peterborough chairman Darragh MacAnthony tweeted: "Just want to wish Dwight Gayle all the best in scaring the life out of Prem League defenders for Crystal Palace. What a signing for them!
"Also want to say that Palace & their Chairman were different class to deal with & incredibly professional. All the best!"