Haveron's superb free came in the 64th minute as he shot into the top corner.
Melaugh netted from close-range on 75 minutes as the ball fell for him when team-mate Stuart King mishit his effort on goal.
Glentoran came close to levelling as David Scullion and Michael Halliday both saw efforts strike the United crossbar.
And, with the Glens piling forward, Kelbie scored a breakaway goal for Ballymena.
It stretched fourth-placed Ballymena's unbeaten run to 11 league matches.
"The main thing has been belief," said Sky Blues boss Tommy Wright.
"During our bad run maybe only myself and the staff believed in the players we had.
"You have to give them credit today as they were behind twice against a good Glentoran team.
"They always believed they could win and it was a deserved victory."