The trade was first reported on Friday and was pending a physical and approval by Major League Baseball.
The Pirates will pay between $13-14 million of the $33 million remaining on Burnett's contract, the Daily News reported Friday.
The Yankees signed Burnett after the 2008 season, around the same time they also landed CC Sabathia.
But unlike Sabathia, who is 59-23 with the Yankees, Burnett struggled to produce in the rotation and finished three seasons for New York with a 34-35 mark.
Burnett, 35, owns a career record of 121-111 with a 4.10 earned-run average in 314 games -- 309 starts -- for the Marlins, Blue Jays and Yankees. He threw a no-hitter while with Florida on May 12, 2001.
"A.J. Burnett is a solid veteran starting pitcher with an above average pitch repertoire and potential to provide us with significant quality innings from our starting rotation," Pirates general manager Neal Huntington said.
Moreno, 24, went 2-4 with five saves and a 3.63 ERA in 41 games between Class- A and Double-A ball last season. Cayones, 20, hit .228 with 12 RBI in 38 total games with State College and the GCL Pirates in 2011.