Rana remained unbeaten on 57 off 44 balls while Rinku made 54 off 43 deliveries to set up KKR's chase of 145 runs after they lost their first three wickets for 33 runs inside the fifth over.
KKR finished at 147 for four in 18.3 overs with Rana leading from the front after making the most of a dropped chance.
KKR's cause was helped to a large extent by a superb bowling effort earlier by spin duo of Varun Chakravarthy and Sunil Narine, who did superbly and picked up two wickets each.
The defeat also puts on hold CSK's chances of sealing a play-off berth as the home side is in second spot with 15 points.
KKR, on the other hand, moved to 12 points with a match to play, but their hopes rest on other results.
Rana and Rinku, both left-handers, showed necessary skills and patience on a tacky pitch to negate the impact of CSK spinners after the home team had posted a moderate 144 for six in 20 overs.
The match-winning partnership between Rana and Rinku ended when the latter was run out by a direct hit from Moeen Ali.
CSK spinners -- Ravindra Jadeja, Maheesh Theekshana and Moeen -- could not do any damage after Deepak Chahar had provided the initial breakthroughs to leave KKR at 33 for 3.
CSK struck an early blow when Chahar got Rahmanullah Gurbaz (1) as he failed to keep his square-drive down and was well caught by Tushar Deshpande close to the boundary.
After six overs, KKR were 46 for three with Chahar having picked up all the three wickets to fall -- Gurbaz, Jason Roy (12) and Venkatesh Iyer (9).
Rana got a life on 18 when Matheesha Pathirana dropped a catch running in from the boundary off Moeen.
In the 13th over bowled by Moeen, Rana played two superb shots on the off-side to pick up boundaries and followed it up with a few more cracking strikes in the 15th over by Maheesh Theekshana.
Earlier, talismanic Mahendra Singh Dhoni's decision to drop down the order backfired as CSK lost wickets at regular intervals to be restricted to the modest score.
Dhoni, who came out to bat no.7 position, remained unbeaten on 2 off 3 balls.
Opting to bat, CSK were 72 for five in in the 11th over before Shivam Dube underlined his utility, hammering a 34-ball 48 (1x4, 3x6s) to help the hosts somewhat recuperate in the company of Ravindra Jadeja (20 off 24).
The duo shared 68-run stand for the sixth wicket to take CSK close to the 150-run mark, which at one point of time looked out of reach on a sluggish pitch.
Spin duo of Narine (2/15) and Chakravarthy (2/36) were the pick of the bowlers for KKR and they were ably supported by Shardul Thakur (3-0-15-1) did and Vaibhav Arora (1/30).
CSK were 99 for five after 16 overs as KKR bowlers, especially the spinners, kept things tight and struck at regular intervals.
Narine and Chakravarthy did the damage in the middle of the innings before Dube and Jadeja launched a fightback.
After a quick start, Chakravarthy broke through for KKR, getting Ruturaj Gaikwad (17, 13 balls, 2x4) to edge one to Arora in the fourth over as CSK raced away to 31 in the fourth over.
Devon Conway and Ajinkya Rahane maintained the pace of scoring, despite not going bang-bang, and scored 52 runs in the first six overs.
Conway earned a reprieve when he got a LBW decision overturned via DRS after being adjudged out off Chakravarthy.
Rahane fell trying to force the pace, unable to clear the boundary off Chakravarthy.
Conway top-edged a pull to Rinku Singh in the deep off Thakur after making 30. Ambati Rayudu's poor season continued as he fell for just 4 to give Narine his first wicket.
The West Indian spinner then castled a struggling Moeen (1) to reduce CSK to 72 for 5 in the 11th over.