The visitors were given a scintillating start in the run-chase when Abhishek Sharma and Travis Head put on 129 for the opening partnership inside nine overs.
Sharma was dismissed for 45 while Australia batter Head went on to score 79 off just 30 deliveries to put the Sunrisers in with a chance.
Klaasen came in at number 4 and was well-supported Nitish Kumar Reddy, who struck 21, and Salil Arora smashed an unbeaten 30 off just 10 balls to take the visitors to victory.
Earlier, Mumbai Indians got to a total of 243-5 thanks largely to an unbeaten 123 by Proteas international Ryan Rickelton while opening partner Will Jacks contributed 46.
It was Rickelton's maiden century in the IPL and it took and cameo down the order from skipper Hardik Pandya, 31 from 15 deliveries, to his team over the 240-run mark.
It was a sixth defeat of the season for Mumbai Indians while Sunrisers got up for their sixth win in nine outings.
Mitchell Starc is set to hand Delhi Capitals a timely boost ahead of their Indian Premier League clash against Rajasthan Royals in Jaipur on Friday, with the left-arm quick cleared to return from May 1.
Legendary spinner Muthiah Muralidaran doesn't see the Indian Premier League striking a balance between bat and ball anytime soon.
Proteas international Donovan Ferreira proved to be the hero with the bat as Rajasthan Royals beat Punjab Kings by six wickets with two balls to spare in their IPL clash.
Josh Hazlewood was the star bowler on show to help the Royal Challengers Bangalore thrash Delhi Capitals by nine wickets in their Indian Premier League clash at the Arun Jaitley Stadium.
