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Power won the prestigious event for the first time after taking the lead from Stefan Wilson with four laps remaining as Wilson didn't have enough fuel to last to the finish.
Power, the 2014 Indy Car series winner, is the first Australian to win the Indy 500. Pole sitter Ed Carpenter was second with Dixon in third.
It was a tense ending to the race with a seventh yellow flag leaving a sprint finish. Oriol Servia, who started in 26th, held the lead at the restart but was quickly passed by Wilson. However both cars were low on fuel and were forced to pit.
That opened it up for Power who led the race for large chunks to retake top spot and eventually win by more than three seconds.
"Man, I just can't believe it," Power said after the victory.
"With one lap to go I was screaming 'man, I got this'.
"I was wondering if I would ever win it," added Power, who finished second in 2015.