Chasing 221 for victory, five‑time champions Mumbai rode a commanding 148-run opening stand from Rohit (78) and Rickelton (81) to reach their target with five balls to spare at the Wankhede Stadium.
Mumbai - the most successful team in the IPL, along with fellow five-time winners Chennai Super Kings - won their opening match in the league for the first time since 2012.
"It's been a long wait - 13 years is a long time," skipper Hardik Pandya said on breaking the opening game jinx.
"Very glad, the whole team is very glad that we were finally able to do it."
The 38-year-old Rohit, who plays just the 50-over format for India after retiring from Test and T20 internationals, hit six fours and six sixes in his 38-ball knock.
His batting blitz came a day after long-time international teammate Virat Kohli steered Royal Challengers Bengaluru to a big win in the tournament opener.
It needed a special effort to dismiss Rohit, and Anukul Roy pulled off a stunning catch while running back from mid-on to finally dislodge the star batter.
South African wicketkeeper-batter Rickelton kept up the attack as he hit his eighth six after Rohit's departure, continuing the pummelling of the Kolkata attack.
He was finally run out after a direct throw from Roy hit the stumps at the non-striker's end.
Mumbai's chase slowed down before Pandya, unbeaten on 18, and Tilak Varma, who made 20, helped the team get over the line.
"I thought 220-225 was a good total. We batted really well, but again, the inexperienced bowling attack," Kolkata skipper Ajinkya Rahane said.
"Playing against Mumbai Indians, strong batting lineup, it's really a learning curve for us."
Pandya won the toss and invited three-time winners Kolkata to bat first amid a loud roar from the home Mumbai crowd.
Rahane and New Zealand's Finn Allen handed Kolkata a blazing start, and the team finished on 220-4.
Rahane went on to hit 67 off 40 balls after Allen fell for 37. Australia all-rounder Cameron Green, the most expensive overseas buy in IPL history, contributed 18.
Rinku Singh's unbeaten 33 off 21 balls gave Kolkata a late lift, but it wasn't enough to challenge Mumbai.
Mumbai medium-pace bowler Shardul Thakur returned figures of 3-39 to be named player of the match, while Jasprit Bumrah finished wicketless but played a role in checking Kolkata's scoring in the middle overs.