Pearse Divine was sent off on the 80th minute while Mark Miskimmin scored the side's only goal six minutes later.
A double from Brian McCaul and strikes by Philip Lowry and Matthew Tipton helped the Blues secure an emphatic win.
They now move to just four points behind second-placed Crusaders in the Danske Bank Premiership, with Cliftonville nine points clear on top.
Elsewhere, Dungannon Swifts and Ballymena United played out a scoreless draw in their clash at Stangmore Park.
Tuesday's Irish Premiership results:
Donegal Celtic 1-4 Linfield
Donegal Celtic: Miskimmin 86.
Linfield: Lowry 2, McCaul 55 (PEN) + 58, Tipton 57.
Dungannon Swifts 0-0 Ballymena Utd