In a game which saw six goals and two red cards, it was Portadown who drew first blood when Matthew Tipton scored a quick-fire double with headed goals a minute apart as Ronnie McFall's side raced into a 2-0 lead by the 16th minute.
The Reds replied just after the break through Diarmuid O'Carroll, before Neil McCafferty received his marching orders in the 72nd minute after a foul on Martin Donnelly inside the area.
Donnelly dispatched the resulting penalty, but the sides were level three minutes later when Tipton fired in to complete his hat-trick.
However, Catney's late strike ensured a share of the spoils, although the visitors had to endure the final seconds without goalkeeper John Connolly, who was given a straight red in the dying moments for handling the ball outside the area.
With both the teams below them taking just a point from their encounter, Linfield moved nine points clear at the top with a 4-1 victory over Carrick Rangers.
Peter Thompson opened the scoring in the 28th minute when he finished off after Phillip Lowry's initial shot came back off the post, but Carrick levelled seven minutes later when Paul Heatley set up Shane Dolan.
Linfield had to wait until the 78th minute to retake the lead, when Jamie Mulgrew's low drive beat Carrick keeper Declan Brown.
Daryl Fordyce added a third two minutes later, before Mulgrew put the gloss on the performance with a strike four minutes from time.
Elsewhere Glentoran suffered a shock 1-0 defeat away to second-from-bottom Glenavon.
Kyle Neill scored the decisive goal with a glancing header early in the second period as the Lurgan Blues recorded a much-needed win.
Over at the Coleraine Showgrounds Curtis Allen was the hero for Coleraine as the Bannsiders avenged last week's Irish Cup defeat against Crusaders with victory over Stephen Baxter's side at the Coleraine Showgrounds.
Allen supplied the finish after David Ogilvy had provided the knock-down from a corner in the 77th minute.
Meanwhile Ballymena United and Lisburn Distillery shared the spoils in a 2-2 draw at the Ballymena Showgrounds.
Alan Davidson, whose goal in the Irish Cup almost cost United their place in the competition after it emerged that he was technically ineligible to play in the competition, netted the opener just before the half-hour mark.
However, the Whites equalised from a fortuitous Mark Cooling penalty - his effort from the spot first struck the post before going in off the head of Ballymena goalkeeper Dwayne Nelson.
Gary McCutcheon netted his 30th goal of the season to restore the hosts' lead, but Distillery made sure of a point when Gary Thompson finished from 6 yards in the 73rd minute.
The game between Donegal Celtic and Dungannon was postponed on Saturday morning due to a waterlogged pitch.
IFA Premiership Results (10/03/12):
Ballymena Utd 2 - 2 Lisburn Distillery
Ballymena Utd: Davidson 29, McCutcheon 42.
Lisburn Distillery: Cooling 34 (penalty), G Thompson 73.
Coleraine 1 - 0 Crusaders
Coleraine: Allen 77.
Glenavon 1 - 0 Glentoran
Glenavon : Neill 46.
Linfield 4 - 1 Carrick Rangers
Linfield: Thompson 28, Mulgrew 78, 86, Fordyce 80.
Carrick Rangers: Dolan 35.
Portadown 3 - 3 Cliftonville
Portadown: Tipton 12, 15, 76 McCafferty sent off 7.
Cliftonville: O'Carroll 53, Donnelly 72 (penalty), Catney 89, Connolly sent off 90.