After trailing by as many as 11 points in the first half, the Suns took their first lead since the opening minutes on a layup by Dudley with 8:03 left in the third quarter before setting for a 63-all tie at the end of the period.
The teams traded baskets to open the final quarter before the Suns took a 73-72 lead on a 15-footer by Jermaine O'Neal with 7:35 remaining.
"When we are solid we can win some games," Dragic said.
Phoenix stretched the lead to 80-76 on a hook shot by Marcin Gortat with 4:05 remaining, but missed a pair of shots, turned the ball over twice and did not score again until Dragic's decisive basket.
"We were playing decent defense and getting stops but offensively we were not executing," Gasol said. "It's hard to win when you don't execute."
NOTES: The Suns have won five games this season after trailing by double digits, all at home. ... Phoenix is 4-3 in games decided by three points or fewer. ... Luis Scola returned to the Phoenix starting lineup after coming off the bench in the team's past 11 games. ... Memphis has not won in Phoenix since Dec. 8, 2010.