The Blues went ahead at Solitude after 29 minutes when Daryl Fordyce, on for the injured Mark McAllister, struck a low drive past John Donnelly, and the Reds' suffered a further setback a minute later when defender Mark Clarke was sent off for a second bookable offence.
Fordyce doubled the visitors' advantage five minutes before the break when he poked the ball home from a Phillip Lowry header, but Joe Gormley gave the hosts hope with a close-range headed goal in the 62nd minute.
However, any thoughts of a comeback by the hosts were dashed when Donnelly could only parry a William Murphy strike into the path of Peter Thompson, who nodded home from close-range to seal a 3-1 win.
Portadown meanwhile suffered a shock 2-0 defeat at Stangmore Park, thanks to goals from Sean Friars and Stefan Lavery.
Friars struck first on the stroke of half-time with a close-range finish from a Mark Brolly cross.
Lavery made sure of the points with a cheeky lob over Ports keeper David Miskelly in the 63rd minute.
Elsewhere Aaron Boyd grabbed a debut goal for Coleraine, but it took a last-gasp equaliser from David Scullion to ensure a share of the spoils for the Bannsiders in a six-goal thriller against Glentoran at the Oval.
Boyd's strike after eight minutes was cancelled out ten minutes later by Stephen Carson, who capitalized on an error by the visitors' goalkeeper Gavin Cullen.
David Scullion restored Coleraine's advantage on the half hour, but Carson fired his second of the afternoon to restore parity early in the second half.
Richard Gibson looked to have won the game with a cool finish in the final minute, before Scullion grabbed an injury-time leveler.
In the game at Mourneview Park, Gary McCutcheon struck twice as Ballymena United earned a 2-0 victory over struggling Glenavon.
McCutcheon, who has announced his intention to join Crusaders at the end of the season, hit the opener in the 31st minute when his effort was deflected in off Jay Magee.
The veteran striker doubled his tally from the penalty spot in the 40th minute after a foul on Johnny Taylor.
Also on Saturday, Lisburn Distillery came from behind to earn a 2-2 draw with Crusaders in a see-saw game at Seaview.
David Cushley opened the scoring for the visitors in style with a fiercely-struck goal 35 yards, before David Rainey pulled one back with a firm header just before the hour mark.
The Crues went ahead in the 70th minute when Timmy Adamson fired home an acrobatic overhead kick after Ryan McCann's penalty struck the crossbar, but the lead lasted just four minutes, with Gary Liggett heading home a Neal Gawley corner.
In Friday night's game Donegal Celtic were 3-2 winners over Carrick Rangers at Seaview.
IFA Carling Premieship Results:
Saturday 25 February:
Cliftonville 1 - 3 Linfield
Cliftonville: Gormley 62, Clarke sent off.
Linfield: Fordyce 29, 40, Thompson 67.
Crusaders 2 - 2 Lisburn Distillery
Crusaders: Rainey 57, Adamson 70.
Distillery: Cushley 8, Liggett 74.
Dungannon Swifts 2 - 0 Portadown
Dungannon: Friars 45, Lavery 63.
Glenavon 0 - 2 Ballymena United
Ballymena: McCutcheon 31, 40.
Glentoran 3 - 3 Coleraine
Glentoran: Carson 18 & 56, Gibson 90.
Coleraine: Boyd 8, Scullion 29 & 91.