Chasing a victory target of 183 runs for victory, Mumbai Indians, led by Harbhajan Singh, could muster only 116 for the loss of nine wickets in their stipulated 20 overs.
Sangakkara's 56-ball 94 that contained 13 boundaries and one six helped him to win the man of the match award.
Australian pace spearhead Brett Lee gave the home team a perfect start as he bowled a fiery spell and accounted for Sanath Jayasurya's wicket before troubling him long enough with a couple of close calls.
Lee bowled at express pace and was very miser with the economy rate as he gave away just nine runs off his four overs.
Besides Lee, Irfan Pathan (2 for 19) and Sreesanth (2 for 31) also denied Mumbai Indians' any liberty to reach anywhere close to their target.
Leg-spinner Piyush Chawla bagged two for 16 while none of the visitors' batsmen had any major contributions with the bat. Robin Uthappa (23) and Dwayne Bravo (23) were the highest scorers.
Both the teams desperately needed to win this game and had lost all their matches before reaching here. But eventually it was Kings XI which managed to register a win and that too a convincing one.
Earlier, after winning the toss and electing to bat first, Harbhajan Singh led Mumbai Indians struck an early blow as Kings XI Punjab opener Irfan Pathan, who opened with Karan Goel, was consumed by medium pacer Ashish Nehra for six runs.
Sangakkara then joined opener Karan Goel and they put on 73 runs for the second wicket. Karan Goel (18) became the victim of a dreadful mix-up when the score was reading 81.
Displaying his form, Sangakkara was seen in full flow and hit Shaun Pollock (1 for 40) and Dhawal Kulkarni (2 for 37) with a lot of ease.
The Sri Lankan's innings was finally terminated by Harbhajan Singh who had him caught by Kulkarni when the left-hander was tried to hit the Indian offie out of the ground.
Yuvraj Singh stepped in amidst loud cheers from the spectators but could not last long. He departed after contributing 18 runs, which included a six and a four.
For Mumbai Indians, off-spinner Harbhajan Singh was the most successful bowler and captured three wickets while Dhawal Kulkarni bagged two wickets for 37. Medium pacer Ashish Nehra too chipped in with a couple of wickets.