The Jambos travelled to Glasgow on the back of eight straight wins but were sent back to the Scottish capital with their tails between their legs after Rangers ran riot.
New signing Mohamed Diomande fired the home side ahead with just over a minute on the clock.
Another January recruit Oscar Cortes doubled that lead after 37 minutes before Cyriel Dessers made it 3-0 before half-time.
Dessers struck again early in the second period before Fabio Silva rounded off the scoring on the hour mark.
A 10th consecutive win for Rangers in all competitions piles the pressure on under-fire Celtic boss Brendan Rodgers ahead of the Hoops' trip to Motherwell on Sunday.
Kilmarnock are next in line for Rangers in midweek but Derek McInnes' side will also go into that game full of confidence after beating Aberdeen 2-0 to strengthen their hold on fourth place.
Corrie Ndaba and Matty Kennedy were on target for Killie to leave Neil Warnock still waiting for his first league win in charge of Aberdeen.
Hibernian closed to within two points of Dundee in the battle for a place in the top six thanks to Myziane Maolida's late strike for a 2-1 win at Easter Road.
Ross County also needed a late winner to beat Livingston 3-2 in a clash between the bottom two.
Josh Sims' 92nd-minute goal took County six points clear of Livie in the relegation battle.
St. Mirren beat St. Johnstone 2-0 to open up a six-point cushion in fifth place.
