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.
Vladimir Weiss impressed with a double for Rangers 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.
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. However, the cheers turned to jeers for referee Craig Thomson when Jelavic was booked for diving just five minutes later.
RANGERS: McGregor, Whittaker, Webster, Weir, Foster, Davis, McCulloch, Ness (Edu 80), Weiss (Fleck 59), Lafferty, Beattie (Jelavic 59).
INVERNESS CT: Esson, Tokely, Innes, Munro, Shinnie, Ross, Duff, Odhiambo, Sanchez (Sutherland 70), Foran, Rooney (Proctor 83).
Ref: C Thomson (Scotland).