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PBKS chased down the target of 163 with 13 balls to spare and notched up their fifth win on the trot against the defenidng champions.
The 26-year-old Rinku failed to make it to the 15-member Indian squad for next month's T20 World Cup.
PBKS spinners Harpreet Brar and Rahul Chahar spun a web around the CSK batters.
However, there is a silver lining for the 21-year-old Delhi man as he is all set to be given a pace bowling contract by the Board of Control for Cricket in India.
Mayank made a comeback in LSG's win against Mumbai Indians in Lucknow on Tuesday, but walked off the field without completing his fourth over after an injury scare.
Mitchell Marsh was confirmed as captain of the 15-man squad which includes Josh Inglis as a reserve wicketkeeper.
The win helped LSG climb to third spot with 12 points from 10 matches, while MI are in ninth place with just six points from 10 outings.
LSG chased down the modest target in 19.2 overs.
The par score was 28 and the Indian team was ahead by 19 runs, and with the surface rendered unplayable by the heavy rain, the visitors were declared winners.
The absence of a cool finisher like Rinku Singh could hurt the India in the big games.