The deadlock was broken in the 16th min when Greenock put in ball from the left which was missed at the front post then travelled without a Carluke challenge to the back post where all that was needed was aa tap in for 1-0 to the visitors.
Carluke created their first chance after 19 mins when Gordon Murphy played an excellent ball to Ryan Smillie and he in turn played an equally good ball to Ian Watt and a great touch saw him control the ball before placing a good shot on target that was well saved by the visiting keeper.
Carluke were playing the better football at this point and in 21 mins they saw another chance end with a corner which ended at the feet of Stevie Murray who poked the ball goal ward but a superb save by the Greenock keeper kept their lead intact.
Jonny Grier in the Rovers goal was called in to action with a fine save in the 25th min from a Greenock free kick following a foul from Murphy.
Chances continued to fall to Rovers and Smillie collected a ball on the right with a wee chance of a volley but he played a cut back after controlling the ball and there were no Carluke takers and another chance was lost.
The final chance of the half saw Greig Cavanagh again hold off his marker on the right and play in a great ball for Smillie but as he tried to switch the ball to his right foot a superb saving challenge ended with only a corner for Rovers.
Half time Carluke Rovers 0 v 1 Greenock FC.
The second half saw Carluke burst out of the traps and they drew level only 30 seconds in to the half when Murphy played a pinpoint pass to Cavanagh who turned and fired the ball for goal but another fine save from the visiting keeper prevented a score but Watt was first to react and struck home a fine equalising goal for 1-1.
If this was good better was to come a minute later when Smillie was clear on the right and cut in to the penalty area only to be upended at the expense of a penalty to Rovers which was buried in the net by Mark Weir for 2-1 to Carluke.
For long spells in the half Rovers looked likeliest to get the next strike with chances falling to Murphy and Cavanagh and Watt having a shot saved after a well worked move but they could not increase their lead.
Greenock showed they were still in the match and brought out another fine save from Grier in the Rovers goal in 71 mins before having their best spell of the second half but they could not find a way past Stevie Murrays defence.
Cavanagh linked up again with Watt in the 79th minute with good close control but a weak shot was easily dealt with by the Greenock keeper.
The last chances of the match fell to Greenock when put on a substitute who almost scored with his first touch of the ball but Grier was on hand with another good save then in the final minute they put in a ball from the right but a back post glancing header went just wide of target.
Carluke now go on to play the winners of Wishaw v Blantyre vics in the next round.
Grier, Lithgow, Murray, Souter, Wilson(Dickson), Kindlan, Murphy, Weir, Cavanagh, Watt, Smillie.