Two-goal Joel Cooper made it another night of misery for Warren Feeney’s team, who were still smarting from their weekend league defeat by Carrick Rangers.
Although Jay Donnelly pulled one back late on, it was too little to late.
But Healy’s men appear to be going from strength to strength. They are now 10 games unbeaten in all competitions and have a healthy eight-point lead at the top of the league table
In contrast, Glentoran’s chances of lifting silverware this season – they trail the Blues by 16 points in the title race – remain now with the County Antrim Shield or Irish Cup.
Following a mundane opening period, Healy’s team broke the deadlock on 14 minutes – the game’s first tangible chance.
Jamie Mulgrew and Chris Shields combined on the left and when the former Dundalk man drilled in a wonderful diagonal ball, Cooper was first to react.
Having somehow escaped his marker, Marcus Kane, Cooper took a touch before drilling the ball low under the diving Aaron McCarey.
Healy’s defensive plans were disrupted on 20 minutes when centre-back Ben Hall sustained a head injury after a collision with Jay Donnelly.
After a lengthy period of treatment, Hall was duly replaced by Josh Archer with Shields dropping back into defence.
It has to be said, the first-half was a low key affair with little goalmouth incident.
The Blues had another half chance just after the half-hour mark when striker Matthew Fitzpatrick was hauled down by Fuad Sule as he cut in from the left.
When Kyle McClean drilled in the free kick, it found the head of Matthew Clarke, but his effort fizzed over the top.
With the fans becoming increasingly frustrated, Warren Feeney’s team did show more conviction as the interval approached.
Daire O’Connor got the better of Clarke on the right and, when he cut the ball back to Donnelly, the striker totally missed his kick in front of a gaping goal.
Then, seconds before the whistle, Sean Murray’s corner kick was helped back into the box by Sule. Although Chris Johns got a telling punch, the ball fell invitingly to defender James Singleton, who could only slice his effort wide.
The Blues struck again only 60 seconds after the restart – and what a beauty it was.
Clarke’s cross from the left was headed straight to Cooper by Russell on the edge of the box and the mercurial former Oxford United player curled the ball over a stranded McCarey.
Now oozing with confidence, the visitors tore the Glentoran defence to shreds on 55 minutes with Millar sending Cooper free on the right.
When he pulled a delicious pass back to Jamie Mulgrew, he ballooned high over the top – a let-off for the home side.
Glentoran were given a rare sight at goal just on the hour when Archer’s misplaced pass was picked up by Donnelly, but he could only shoot feebly wide.
The action quickly switched to the other end and impressive defender Euan East was right out of luck when he met a Cooper corner kick only to see his header flash inches over.
Cooper almost grabbed his hat-trick 16 minutes from the end when he met a cross from substitute Chris McKee only for McCarey to pull off a wonder save as the ball hurtled towards the top corner.
Against all the odds, the Glens hauled themselves back into the tie seven minutes from time when O’Connor’s cross was headed home by Jay Donnelly.
Seconds later the Blues should have been out of sight when Millar found McClean, who blazed a great chance over the crossbar from 10 yards.