The Reds were in impressive form at Shamrock Park, and took a deserved lead as early as the eighth minute when Diarmuid O'Carroll arrowed a shot past David Miskelly from a narrow angle.
Joe Gormley doubled the visitors' advantage midway through the second half when he found the corner of the net in the 65th minute from George McMullan's free kick.
Liam Boyce continued his scintillating form this season when he capitalised on a defensive blunder and struck a superb curled effort into the net for Cliftonville's third goal with five minutes left.
Elsewhere Crusaders kept up the pressure on the league leaders as they thrashed Ballymena United 5-1 at Seaview.
The Crues grabbed three goals inside five first half minutes to assert their dominance over the Sky Blues, with Gary McCutcheon scoring the first from a Paul Heatley pass in the 36th minute
Heatley got on the score-sheet himself two minutes later when he pounced after Dwayne Nelson saved another shot from McCutcheon, before a mistake by the Ballymena stopper allowed Chris Morrow to chip the ball into the net from all of 40 yards.
Colin Coates got in on the action when he fired home a trademark thumping header from a Heatley corner in the 60th minute, before McCutcheon added his second of the afternoon 11 minutes from time.
Allan Teggart netted a late consolation for the visitors, but Stephen Baxter's side were full value for their victory, and move seven points clear of third-placed Linfield, whose game against Donegal Celtic was postponed due to the weather.
Elsewhere Ballinamallard made it two wins on the bounce with a 3-1 victory at home to Glenavon.
The Mallards opened the scoring inside the first minute, with Steven Feeney blasting home from Ryan Campbell's header, and Leon Carters added a second six minutes before the break when he fired in from Jason McCartney's corner.
Feeney added his second of the afternoon when he fired in on the rebound in the 67th minute after Chris Currans initial effort had been saved by Andy Coleman.
Alvin Rouse did well to keep out Kyle Neill's 70th minute penalty, but he was eventually beaten when Guy Bates grabbed a consolation for the visitors with a diving header three minutes later.
Finally Coleraine returned to winning ways with a 3-1 victory over struggling Lisburn Distillery.
Stephen Lowry opened the scoring with a powerful header in the 16th minute, before Ruairi Harkin finished with ease after a sweeping move by the hosts in the 23rd minute.
Kyle McVey headed a third goal for the Bannsiders just before the break, but Owain Beggs gave his side hope when he raced through to beat the offside trap and slot home for the Whites.
However it wasn't to be for Tim McCann's side, who remain rooted to the foot of the table.
Wintry weather conditions meant two of Saturday's fixtures were called off.
Donegal Celtic's home game against Linfield was called off after a pitch inspection on Friday.
Glentoran's game against Dungannon Swifts at the Oval was postponed due to a waterlogged pitch caused by melting snow.
Saturday's Danske Bank results:
Ballinamallard United 3-1 Glenavon
Ballinamallard Utd: Feeney 1, 67; Carters 39.
Glenavon: Bates 73.
Coleraine 3-1 Lisburn Distillery
Coleraine: Lowry 16, Harkin 23, McVey 45.
Lisburn Distillery: Beggs 46.
Crusaders 5-1 Ballymena United
Crusaders: McCutcheon 36, 69; Heatley 38, Morrow 40, Coates 60.
Ballymena Utd: Teggart 79.
Portadown 1-3 Cliftonville
Portadown: Mackle 75
Cliftonville: O'Carrroll 8, Gormley 66, Boyce 85.