Late in the first half Glentoran had Jonny Taylor sent-off but McCafferty's resultant penalty was saved.
Glentoran protested and referee Moutray consulted assistant Gareth Eakin before confirming the spot-kick verdict.
However, Dungannon striker Morgan said Glentoran's Sean Ward had handled the ball just before he did.
The goal keeps Linfield two points behind Glentoran and the Blues have a game in hand.
Cliftonville remain five points behind leaders Glentoran but the Reds have a game in hand.
Fordyce, restored to Glentoran's starting line-up, stroked in Glentoran's fourth with Gawley shooting in the fifth.
Another Glentoran goal would have had strained Linfield nerves and the chances did come late on.
The Blues saw off challenges from Glentoran, Portadown and Derry City to land the highly-rated midfielder.
Ciaran Martyn doubled Glentoran's advantage on 26 minutes and Fordyce made it 3-0 on 44 minutes.
Fordyce got Glentoran's second in the 51st minute with a great shot from Fitzgerald's pass.
BBC: Glentoran beat Linfield on away goals in quarter-finals
Elliot Morris denied Stephen Carson before Jonny Black volleyed Glentoran's winner 18 minutes from time.
The win moves Glenavon up to second place in the table - one point behind leaders Glentoran.
Glentoran almost gifted the hosts an equaliser with James Taylor punching a cross onto his own crossbar.
Northern Ireland's Glentoran have a tough tie against Norwegian outfit SK Brann and Portadown face FBK Kaunas.
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The Reds striker topped the poll ahead of second-placed Jimmy Callacher of Glentoran and Glenavon's Mark Haughey.
League leaders Glentoran will be seeking a third straight win when they travel to take on Newry City.
In the first half Glentoran's Johnny Black shot wide while Colin Nixon headed over from a Black corner.
BBC: Glentoran beat Linfield on away goals in quarter-finals
Right at the death, Leeman cleared off the line as Glentoran held on to take the three points.
Crusaders' win moves them level on points with second-place Glentoran with both teams five points behind leaders Linfield.
Two IFA Premiership football games - Glentoran v Dungannon Swifts and Donegal Celtic v Linfield - were postponed.
Michael Halliday, who has been released by Glentoran, and Richard Gibson are among the players who may join United.
Carson's long-range effort then tested Morris before Black hit Glentoran's winner on 72.
Afterwards, Glentoran's former Northern Ireland star Keith Gillespie said they had been amazed when the penalty was awarded for handball.
Gillespie completed a shock move to Irish League champions Glentoran last week and said that his international ambitions remained alive.
Referee Trevor Moutray awarded the spot-kick for handball despite adamant Glentoran claims that a Dungannon player had been the guilty party.
Glentoran are three points behind the leaders but they have a game in hand on Cliftonville and a better goal difference.
Across the city, Glentoran 's fortunes seem to have changed a bit with a fresh face coming into the manager's chair.
Lord Glentoran insisted he was "not out to knock bicycles", and told peers he recognised the health benefits associated with cycling.
Irish Premiership champions Glentoran plan to parade their star signing at the Oval on Tuesday night before the derby clash with Linfield.
Glentoran kept up the pressure and substitute Gary Hamilton nearly got on the end of a dangerous cross nine minutes from time.
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