The Suns needed a 16-1 run late in regulation to force overtime, and they took care of business once they got there.
P.J. Tucker's trey at 5:36, Phoenix's first made 3-pointer of the game, of the fourth quarter ignited the late run. Tucker scored seven of nine Phoenix points around a Deron Williams free throw to get the Suns within seven.
Bledsoe and Brandon Knight both made two free throws, and Marcus Morris finally hit from deep after missing his first eight attempts, to tie the game at 92. Jarrett Jack's last-second jumper from the top of the key rimmed out.
Neither team led by more than two in the extra session until Knight followed Marcus Morris' jumper with a difficult fadeaway jumper with 17.5 seconds left. Bledsoe and Brandan Wright, who finished with 14 points without missing a shot, both made two free throws to seal the win.
Knight finished with 12 points and seven assists and Tucker, who did not start because he missed the team bus to shoot-around earlier in the day, gave 11 points off the bench.
Brook Lopez led five Brooklyn reserves and six players overall in double figures with 19 points and 13 rebounds.
Phoenix went into halftime with a 50-46 edge, but a late third-quarter run helped Brooklyn jump out to a big lead.
Trailing by one with 2 1/2 minutes left in the period, Bojan Bogdanovic caught fire and hit 3-pointers on three straight possessions. Jack then drove in for a layup and Cory Jefferson had a three-point play to give the Nets an 80-69 lead after three.
Lopez opened the fourth with a layup, and Brooklyn led by 15 points before the Suns rallied back.
The Suns shot 3-for-26 from behind the arc ... Phoenix won its second consecutive road game for the first time since Jan. 6-7 ... Williams finished 2-for-16 from the floor for nine points in 44 minutes, while Jefferson posted 12 points and 13 rebounds in just under 16 minutes ... There were 15 ties and 20 lead changes.