The in-form Yusuf Pathan (37 off 18) helped KKR make short work of the Duckworth-Lewis target of 66 runs. The hosts had nine overs to get home but Pathan ensured they raced to the finishing line in just five overs.
Pune had earlier struggled to 103 for six when rain stopped play for a little more than two hours.
The win pushed KKR to second spot in the standings with 14 points from 11 games. Pune, who are out of the competition, remained at the bottom of the table.
Uthappa was stumped on a viciously turning ball that ended up being a wide on the leg side while Gambhir was foxed by a straighter one off Ashwin.
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In walked Pathan and tilted the game in his team's over by hammering Ashwin for two sixes and as many fours in his second over, also settling the nerves in the KKR dug out. Eventually, he got the team home comfortably in the company of Manish Pandey (15 off 10).
Legspinner Piyush Chawla claimed 2/21 with 13 dot balls while offie Sunil Narine, returning after two matches, gave away a miserly 10 runs from 3.4 overs bowling 14 dot balls to win the spin battle against MS Dhoni and Co.
With KKR spinners making full use of the conditions, the otherwise devastating Dhoni crawled to eight runs from 22 balls, as his counterpart Gautam Gambhir relished the challenge, deploying four fielders around the Indian skipper.
Opting to bat on a turning surface, Pune failed to get going at the top with Russell giving the first breakthrough in the third over.
(REOPENS DEL 51)
Aussie duo of Usman Khawaja and George Bailey perished to spinners after Gambhir introduced Shakib in the fifth over.
Khawaja (21 from 17 balls; 3x4) was playing freely but the Bangladeshi leftarm spinner dismissed the Aussie in his first over when the lefthander smacked straight to Suryakumar Yadav at square-leg.
In between rookie medium pacer Ankit Rajpoot, playing his first match this season, removed Saurabh Tiwary as Dhoni promoted Irfan Pathan at No 5 to step up their run-rate.
But the Knights spin trio revelled in the conditions.
An interesting Gambhir versus Dhoni battle came to fore when the KKR skipper brought on two slips, a short leg and silly point against the Indian captain and added a fifth fielder at short cover with Chawla bowling a tidy spell.
Dhoni struggled to keep the flow and to add to his misery his batting partner Irfan was no match to his running.
As it turned out, Pathan became the victim of a runout when he sacrificed his wicket for Dhoni with utter confusion taking place between the two.
Both teams made two changes with Dhoni giving Irfan his second match of the season while a fit-again Murugan Ashwin also returned after two matches, replacing Rajat Bhatia and RP Singh.