The 32-year-old, who has made 164 appearances since joining the Reds in 2006, hopes to resume pre-season training after suffering a serious shoulder injury in November that ruled him out for the remainder of the season.
"I'm delighted," Dunbavin told the club's official website. "Obviously last year was full of ups and downs personally.
"The vein of form I was in was the best I've ever played I think, not only for Accrington but in my career, so I was devastated to learn that I was going to miss the rest of the season with the injury.
"It's been a case of just getting on with it, having the operation and getting back to being around the lads and giving them my full backing really.
"It's an honour to play for the club and I'll carry on doing so to the best of my ability."
Dunbavin joins George Miller in committing his future to Stanley after the midfielder signed a new deal last week.
Accrington are currently without a manager following Leam Richardson's decision to team back up with former Reds' boss Paul Cook at Chesterfield.