Kolkata kept Chennai down to 144-6, a total they overhauled riding on a key stand of 99 between Rinku (54) and Rana (57 not out) in Chennai.
Spinners Varun Chakravarthy and Sunil Narine took two wickets each to set up the victory as Kolkata jumped a spot to seventh in their attempt to finish the league stage in the top four for a playoff berth.
"I said at the toss that if we do well in all the three departments, then our chances are good," Rana said after his team's sixth win in 13 matches.
Four-time winners Chennai remain second in the 10-team table led by holders Gujarat Titans and on course for the playoffs.
See the full scorecard on Flashscore.
Kolkata looked in trouble at 33-3 after Chennai pace bowler Deepak Chahar's third strike to send back Jason Roy, before the left-handed pair of Rinku and Rana guided the team home with nine balls to spare.
Rinku, who became an IPL sensation with his five sixes off the last five balls to conjure up a miracle win for Kolkata last month, struck his third fifty of the season before being run out. He was named man of the match.
Rana made most of a dropped catch on 18 to reach his fifty and stood firm to hit the winning runs.
Four-time winners Chennai started strongly but slipped from 61-1 to 72-5 in the space of 19 balls before Shivam Dube revived the innings with his unbeaten 48.
Dube kept up the fight in his 34-ball knock laced with three sixes but it was superstar and skipper M.S. Dhoni who brought the house down in his team's last league game at M.A. Chidambaram Stadium.
Dhoni, who many suspect is playing his final IPL, walked into a deafening roar as he faced four balls including a wide and no-ball on which he was bowled but ended on two not out.
