Rangers looked the hungrier side throughout the afternoon and deserved their win with goals from Sone Aluko (11), Andrew Little (72) and Lee Wallace (77) giving them a 3-0 lead after Celtic's young team had been depleted by red cards to Cha Du-Ri and Victor Wanyama in either half.
Celtic manager Neil Lennon was sent to the stands at the break apparently after arguing with referee Calum Murray, but he watched his team rally late in the game with Scott Brown netting a penalty after home defender Carlos Bocanegra was sent off on 88 minutes for a foul on Georgios Samaras as he was set to pull the trigger.
Visiting skipper Scott Brown slipped the penalty into the net before Thomas Rogne powered home a header from Kris Commons's delivery, but there was no time left for the visitors to rescue a point.
The result means little to the destination of the Scottish Premier League title with Celtic holding an 18-point lead over their city rivals with seven games remaining, but Rangers will be happy to prevent Celtic from winning the championship at their stadium.
Celtic could yet win the title if results fall favourably for them before they host St Johnstone at Celtic Park next Sunday. If Rangers win at Motherwell on Saturday, Celtic cannot win the SPL next weekend.
Cha managed to prevent Aluko from getting a shot at goal on four minutes with a fine tackle before debutant Rhys McCabe dragged a shot wide of the target.
Celtic were looking flat and paid the price for being flat-footed as Aluko left Charlie Mulgrew and Rogne for dead before dancing into the area to slip the ball beyond visiting goalkeeper Fraser Forster.
Rangers goalkeeper Allan McGregor made excellent saves from Samaras and Anthony Stokes before Celtic were reduced to 10 men when the referee decided Cha had denied Wallace a goal-scoring opportunity on 28 minutes.
They went down to nine men when Wanyama was sent off 10 minutes after half-time. The Celtic man won the ball from a tackle on Steven Whittaker, but referee Murray deemed him to have gone in dangerously with two feet.
Rangers moved two goals clear when substitute Little stabbed the ball home from close range after Forster had saved a shot by Wallace, who seemed to be offside only for the match officials to miss it.
Wallace added a third goal moments later when Celtic lost possession and he raced onto a Steven Davis ball to beat Forster.
Celtic rallied late in the day with their two goals, but the 3-2 scoreline probably flattered them as they continued their patchy run of recent form a week after losing 1-0 to Kilmarnock in the Scottish League Cup final.