The win allowed the Scottish champions to move to within two points of their rivals at the summit, with two games in hand.
The other highlight of the night was the return of Nikica Jelavic after almost four months out through injury - just as fellow striker Kenny Miller edged closer to sealing his departure.
Miller was missing from the squad amid reports he was set for a move to Bursaspor after Birmingham confirmed he rejected a switch to the Midlands in favour of an "irresistible" offer from the Turks.
Rangers were boosted by the news that Steven Whittaker passed a late fitness test following concerns over a bruised toe but Sasa Papac missed out with a groin strain.
Madjid Bougherra was suspended, paving the way for Andy Webster to make only his second appearance of the season and the first in the SPL.
Inverness made two changes from their goalless draw with Motherwell, drafting Eric Odhiambo and Danni Sanchez into the side.
Rangers tried to apply early pressure and managed to carve out the first real chance when Kyle Lafferty met a Davis cross in front of goal but the opportunity was squandered when the striker flicked the ball over the crossbar.
At the other end, Inverness were threatening when Graeme Shinnie picked out Richie Foran at the back post from a corner and his header was touched onto the crossbar by Allan McGregor before the danger was eventually cleared.
Vladimir Weiss impressed with a double at the weekend and could have found the net again in this game when the clearance from his free-kick fell back to his feet only for the winger to see his lofted effort fall just over the crossbar.
Weiss had another crack at goal, this time with a shot from distance that whistled inches past the upright. He was then the victim of a challenge which earned Odhiambo the first booking of the game.
The breakthrough finally came on the stroke of half-time when Lafferty saw his shot blocked in the box amid claims for handball before the ball broke for Davis to rifle home from the edge of the box.
Inverness tried to haul themselves level again after the restart and Shinnie punted a dangerous ball into the box that was punched to safety by McGregor before Rooney could connect in front of goal.
Ibrox exploded in rapturous applause when Jelavic made his long-awaited comeback from injury with an hour gone as he was thrown into the action with John Fleck, with James Beattie and Weiss both withdrawn.
However, the cheers turned to jeers for referee Craig Thomson when Jelavic was booked for diving just five minutes later after going to ground in the box under pressure from Chris Innes.
Rangers continued to push forward in search of another goal and Shinnie was called into action to clear a Lafferty effort off the line.
Jelavic came close to marking his return with a goal when his stinging drive flashed across goal and narrowly past the post.
The Croatian forward was clearly still well short of sharpness and he passed up another decent chance, this time ballooning over after meeting a cross from the left from Fleck.
Inverness piled forward in numbers in the final minutes but Rangers were able to see out the remainder of the game and claim maximum points.