Law notched either side of a goal from former Killie striker Kris Boyd in a comprehensive fourth-round upset.
Killie were poor throughout the game and were knocked out by their Championship opponents 3-0, with midfielder Law opening the scoring after 18 minutes and adding an excellent third in the final minutes of the match.
The home side gave a clear indication of their attacking intentions from the first whistle, with Lewis MacLeod finding space for a left-footed shot which he dragged wide of the far post.
Rangers were dominant throughout the first half, with Alexei Eremenko’s deflected shot all Kilmarnock had to show for their efforts.
There was a huge appeal for a penalty when the ball bounced up off Tope Obadeyi’s hand but moments later Rangers had the lead, Nicky Law’s shot from 20 yards beating goalkeeper Craig Samson with the aid of a deflection.
MacLeod then produced a nice touch and volley, which Samson took at the second attempt, before the winger played in Kenny Miller only for Ross Barbour to deny him with a sliding, last-ditch tackle.
Daly spurned a good chance to put the game further out of Killie’s reach in the final minutes of the first half but his sidefooted finish was tame and easily held by Samson.
The second period of the game was a quiet affair, with MacLeod’s wild volley the only attacking action of its first 15 minutes.
Jon Daly led a counter-attack as Rangers sought for a result-sealing second goal but did not have the pace to get away from the defence and his attempt through-ball to Miller was sloppy and easily cut off.
Ex-Kilmarnock striker Boyd then came off the bench to rattle in a loose ball at a corner kick to make it 2-0 to Rangers and put the game beyond his former club.
Another substitute, Nicky Clark, almost got in for a third but defender Mark Connolly used his physical presence to ease Clark off the ball and deny him space for a shot.
Rangers’ third came from a well-executed counter-attack down the left wing which ended with Law curling a shot past Samson’s left hand.
It was no less than the home side deserved for such a dominant performance.