The Team Australia driver had a time of 1 minute, 34.9451 on his final lap on the 2.8-mile, 14-turn circuit. Dixon, the Indy 500 winner, had his fastest time on the previous lap, but Power was 0.82 seconds quicker to take the pole for Sunday's race.
"We've been really fast all week," said Power, who led both practice sessions Friday. "I just hope we can hold it all together tomorrow."
Australian Ryan Briscoe was third, followed by Franchitti, who is making his return to open-wheel racing from an aborted stint in NASCAR.
The 18th version of the race Sunday also is the first under the Indy Racing League banner, and is a non-points event. The IRL and Champ Car World Series, which formerly staged the Surfers Paradise event, unified in February.
Under the IRL qualifying format, the 24 cars were divided into two groups in sessions lasting 20 minutes each. The top six of each of those groups advanced to a second round of qualifying of 15 minutes, and the top six from that group went into a final 10-minute session to determine the pole winner.
The first group that included Danica Patrick had light rain falling when the session started. Patrick's bad run in her Australian debut continued when she crashed into the wall on her first lap after managing only a best time of 20th place in any of the practice sessions.
Patrick, the first woman to win a major open-wheel event, finished with no time and will start from the back of the grid Sunday.
Hunter-Reay was the best in the drizzle in the opening qualifying group but 19 seconds slower than the top practice time set earlier in the day on a dry track.
"It was pretty messy out there," said Briscoe, who just sneaked into the second round of qualifying by finishing sixth in the first group.
The rain eased slightly for the second group but the wet track left cars slipping and sliding all over the track and the yellow caution flag came out three times during the session. Power, on his last lap, finished with the fastest time in the second group, 1.45 seconds quicker than Hunter-Reay.
With more showers approaching the track, officials started the next session, with the 12 fastest cars, early and the track dried for most of the session. Dixon led the group of the final six qualifiers, followed by Castroneves, Franchitti, Briscoe, Power and Hunter-Reay.