Mathew Barzal, Kyle Palmieri and Casey Cizikas also scored, and Noah Dobson had two assists to help the Islanders improve to 6-0-1 in their last seven games. Semyon Varlamov made 20 saves.
The Islanders are 9-1-4 in their last 14 games and trail the first-place New York Rangers by just four points in the Metropolitan Division.
Troy Terry, Adam Henrique and Sam Carrick scored for Anaheim, which has lost four straight and 12 of 13. John Gibson finished with 30 saves.
Carrick scored at 5:07 of the third with Anaheim’s third straight goal after New York took a 2-0 lead in the second period. It was Carrick’s sixth of the season.
Barzal tied the score 3-3 on a one-timer off a pass from Bo Horvat at 7:46. It was his 10th goal of the season and gave him a team-leading 30 points.
Holmstrom got the tiebreaking goal with just under 1½ minutes to go, scoring his eighth goal just 12 seconds after defenseman Robert Bortuzzo was sent to the penalty box for tripping.
Palmieri scored 40 seconds into the second period to get the Islanders on the scoreboard first with his ninth.
Cizikas made it 2-0 at 6:32, finishing a give-and-go with Cal Clutterbuck. It was his fifth.
Terry scored his sixth at 7:47 to get the Ducks on the scoreboard and Henrique notched his sixth with 6:31 remaining in the middle period to tie the score 2-2.
Former Islander Ross Johnston received a warm ovation during a video tribiute midway through the first period. Johnston played parts of seven seasons with the Islanders before the Ducks claimed the left wing off waivers in October.