Sharma shared an unbroken fourth-wicket stand of 131 in 14.4 overs with Corey Anderson (55 off 41) to propel Mumbai to 168-3 off their 20 overs.
But it wasn???t enough as Kolkata cruised to 170-3 off 18.3 overs on the back of 57 from Gautam Gambhir and cameos from Suryakumar Yadav (46no off 20) and Manish Pandey (40 off 24).
Mumbai were in trouble early on after being put into bat as South African seamer Morne Morkel (2-18) forced Australian opener Aaron Finch to hole out in the deep of a leading edge for just five.
Morkel, who bowled his four overs straight through, followed up with the wicket of Ambati Rayudu to make it 37-3 after the expensive Shakib Al Hasan had removed Aditya Tare for seven.
But captain Sharma rescued the innings, powering his way past 50 as he struck 18 off four deliveries from Umesh Yadav.
Anderson enjoyed a life in the following over when he was dropped on 23 by Andre Russell at midwicket and Rohit also benefitted from a slice of fortune when he was dropped behind the wicket by Robin Uthappa on 70 off the bowling of Sunil Narine, who returned 0-28 on his return to cricket with a new action.
Knight Riders made light work of chasing down their target, however, even after Anderson had Uthappa caught at slip on the drive for nine.
Gambhir was dropped by Tare off Vinay Kumar before he scored and went on to strike seven fours and a six to keep his team up with the rate before Pandey and Yadav ??? who struck three and five maximums respectively ??? cut loose.
Jasprit Bumrah bore the brunt of the assault, conceding 20 runs off the 17th over of the innings before Yadav wrapped up the victory with a sweetly-timed four off Vinay Kumar.
Reflecting on Kolkata???s victory, Sky Sports pundit Shane Warne said: ???They always had it under control. When you are only chasing a par score you know you can catch it up if you have a bit of a lull. To win with nine balls to spare is a smashing.???
Fellow pundit Michael Vaughan was particularly impressed with the flair batting of Yadav and Pandey.
???That???s what the IPL brings,??? he said. ???The opportunity for young Indian players to get in those dressing rooms and get on the big stage in front of 80,000-90,000 people.
???Their skill levels have to be high but I think it???s the pressure of the actual moments that these guys are being put under that will stand Indian cricket in good stead.???