Dungannon Swifts looked bright from the off and passed up a great chance to open the scoring inside the first 10 minutes as Sean McAllister's inviting delivery across the face of goal eluded everyone in a Swifts' shirt.
Kealan Dillon fizzed a low shot from distance narrowly past the post before Jamie Glackin forced a smart save out of Andrew Mills at his near post following good work from Leo Alves as the Swifts continued to impress.
At the other end, Jordan Stewart looked Glentoran's biggest attacking threat and he should have done better when presented with an opportunity by Ross Clarke.
The pair combined again midway through the half when Stewart played in the former Crusaders man, but he blazed well wide of the target from a good position.
Stewart's influence did eventually tell as the hosts broke the deadlock five minutes before half-time.
The winger got the better of his marker Shea Gordon before delivering an inviting cross into the heart of the Dungannon Swifts penalty area where James Singleton rose highest to guide his header past Alex Henderson for his first goal of the season.
The hosts picked up where they had left off following the restart, but it was Dungannon Swifts who came closest to scoring, midway through the second half.
First, Bigirimana forced a fine fingertip save out of Andrew Mills with a fierce drive from just outside the penalty area before the woodwork came to the Glens' rescue with Dillon's volley crashing off the crossbar from the resulting corner.
Leo Alves forced Andrew Mills into a comfortable save at his near post while, at the other end, Alex Henderson got down well to stop a Jordan Jenkins effort.
The three points were finally secured in added time as Malone raced clear of the Dungannon Swifts defence following a free-kick at the other end to calmly chip Henderson for his second goal of the season.