That margin had grown to 15 points 90 seconds into the second quarter, but from that point, Melbourne started running down and then overtaking the visitors.
The hosts were ahead 46-45 at halftime and a 14-0 run in the third quarter put them in control.
Down 80-68 at the last break, victory looked beyond the Breakers but they worked hard to narrow the gap and a Corey Webster three-pointer with 10 seconds remaining levelled the scores at 96-96 and forced the match into overtime.
The visitors struggled for traction in overtime and Melbourne eased away to a 111-102 victory.
Shawn Long was immense for the Breakers, finishing as the game's top scorer and rebounder with 28 and 15 respectively while also contributing six assists, a steal and a block.
Long was backed by 18 points from Shea Ili, 16 from Webster, 15 from Wesley and 11 from Finn Delany off the bench.
The New Zealand club's final four games are all against other sides vying for the final playoff spot.
The Breakers (10-14) host the fourth-placed Adelaide 36ers (13-12) at Auckland's Spark Arena on Friday night, followed by two games against Illawarra Hawks (11-13) before finishing the regular season against the Brisbane Bullets (12-13).