Kings would have been confident of victory having scored a daunting 245-6, with captain Shreyas Iyer striking 82 from 36 balls.
Prabhsimran Singh and Priyansh Arya got their side off to a strong start with an opening partnership of 66, before Iyer added 71 with Nehal Wadhera.
Iyer was the last of four wickets to fall to Harshal Patel (4-42), while Marcus Stoinis muscled an unbeaten 34 from 11 to close the innings.
But Abhishek and Travis Head got Sunrisers off to a flying start in their reply, passing 100 in the eighth over.
Head was finally caught at long-on by Glenn Maxwell off Yuzvendra Chahal for a 37-ball 66, by which time Sunrisers were 171-1 in the 13th over.
Abhishek reached a 40-ball hundred from the final ball of the same over, raising a piece of paper saying “This is for the Orange Army” in acknowledgement of his side’s supporters.
When Abhishek was finally dismissed, his side were 222-2, and Heinrich Klaasen and Ishan Kishan guided them home with nine balls to spare.
Sunrisers rise two places from the bottom of the table to sit eighth with two wins and four defeats, while Kings remain in sixth.
