Rohit, back after missing five matches with a hamstring injury, was ably supported by Rickelton, whose explosive 83 off 32 balls laid the foundation for Mumbai's chase at the Wankhede Stadium.
Opting to bowl first after winning the toss, Mumbai were set a daunting target as Lucknow posted 228 for five.
Opener Mitchell Marsh made 44, while Nicholas Pooran produced a blistering 63 off just 21 balls. Himmat Singh added a valuable 40 to keep the pressure firmly on the home side.
Mumbai's response was immediate, as openers Rohit and Rickelton put on a commanding partnership.
Rohit struck six fours and seven sixes, while South Africa wicketkeeper-batter Rickelton hit six fours and eight sixes, taking full advantage of the batting-friendly conditions.
Rickelton was dismissed by Mohsin Khan and Rohit fell to M Siddharth, but Mumbai never lost control.
Tilak Varma and Suryakumar Yadav steadied the innings before late contributions from Naman Dhir and Will Jacks guided Mumbai home with eight deliveries to spare.
Yadav captained the side in the absence of regular skipper Hardik Pandya, who was sidelined after feeling unwell.
Rickelton was named Player of the Match and praised his opening partner.
"One of Mumbai's greatest, one of India's greatest. I have enjoyed batting with him," he said.
"My game suits this ground... Really enjoyed it - the ball really flies here. Always looking to get better. The power we have is through the roof. With the class we have, it makes my role clearer."
Mumbai had struggled in Rohit's absence, but his return proved transformative as the hosts completed their highest successful run chase in IPL history. Lucknow, meanwhile, slumped to a sixth straight defeat.
Five-time champions Mumbai remain ninth in the table with six points from 10 matches, while Lucknow are bottom with four points in the 10-team competition. The top four teams in the unified points table qualify for the play-offs.