Power, who dominated last year's Honda Indy Grand Prix of Alabama, this time had to wait until after the final round of pit stops to grab the lead, then needed to fight off Scott Dixon after a restart with 16 laps remaining to secure the win.
The Australian Team Penske driver was denied a spot in the race's "Fast Six" when Ryan Hunter-Reay spun around and stopped on the track during the third qualifying run.
Dixon, meanwhile, settled for a second straight second-place finish, ending 3.37 seconds behind Power.
Pole sitter Helio Castroneves came in third, followed by Graham Rahal and rookie Simon Pagenaud.
Notable deaths of 2014 [PHOTOS]