Doug Weight had a pair of assists for the Islanders to become the eighth American-born player to reach the 1,000-point mark. Zbynek Michalek had a goal and an assist, and Viktor Tikhonov and Joakim Lindstrom each added unassisted goals for the Coyotes, who won for the third time in four games.
Michalek snapped a 1-all tie on a shot from the point with 10 seconds left in the first period.
DiPietro, making just his fifth start of the season and first since December 26, made 28 saves for the Islanders. New York's No. 1 goalie returned from a 27-game absence a week ago and played two games before being sidelined with a groin injury.
Jokinen's goal on a long rebound into the low slot extended Phoenix's lead to 4-2 with 16:50 remaining.
Bailey beat Mikael Tellqvist on a one-timer from between the hashes 31 seconds later to pull the Islanders to 4-2.
Lindstrom stole the puck from the Islanders' defence at the New York blue-line and beat DiPietro on the breakaway with 6:27 to go for his fourth goal in 12 games.
Weight fed Park on a long outlet pass to make it 5-3 with 3:05 left. Park struck again with 28 seconds to go on a one-timer from the bottom of the left circle after DiPietro had gone to the bench for an extra attacker.
Tikhonov opened the scoring when he gained control of the puck in the right faceoff circle, skated to the front of the net and backhanded a shot between DiPietro's pads just 1:46 into the game.
Bergenheim tied the game at 1 at 9:20 of the first on a wrist shot from the top of the left faceoff circle.
Notes: Weight has 16 goals and 46 assists in 58 career games against the Winnipeg/Phoenix franchise, his third-highest total against any single team. ... Doan and Jokinen have scored in the same game three times this season. ... Guerin has assists in four straight games and has scored in five in a row overall. ... Michalek was playing in his 300th career game. ... Steven Reinprecht's assist on Doan's goal was the 200th of his career. ... DiPietro had missed the past three games with a strained groin. ... Jackman caught Tellqvist across the mask with his stick after a whistle in the first period and was booed lustily the rest of the game every time he touched the puck.