Chasing a modest 161-run target, the visitors received a couple of early jolts before Rahul (55) and Sarfaraz (67) anchored the Punjab innings with a crucial 110-run stand for the second wicket. However, it was not enough to help them cross the line as Chennai registered their fourth win from five matches.
With this victory, CSK have jumped up to the top spot on the points table with 8 points in their kitty.
Harbhajan was the best bowler for the hosts with the figures of 4-1-17-2. In a brilliant spell, he bowled a maiden and grabbed two wickets, including that of the 'universal boss' Chris Gayle in the second over.
Two balls later, he had the stylish Mayank Agarwal, caught by Faf du Plessis near the boundary with the batsman going for a big hit. Harbhajan started off with a double wicket maiden to put Kings XI on the back foot early.
In the 13th over, Rahul's missed sweep saw Dhoni flick the ball on to the stumps but the bails didn't come off through the LED lights lit up, ensuring the batsman survived despite being short of his crease.
Ravindra Jadeja (4-0-24-0) and Imran Tahir (4-0-20-0) also helped CSK put the brakes on the rival batsmen. They, along with Harbhajan, conceded only 61 runs in 12 overs even as the required run-rate kept mounting.
Kuggeleijn did not start very well, being hit for a six off his very first ball, but did well to finish with 2 for 37 in his 4 overs.
Earlier, it looked like smooth sailing for CSK with the score reading 98/1 in the 13th over. But Punjab skipper Ravi Ashwin had other ideas.
CSK looked comfortably placed, having scored 54 runs in the Powerplay without losing a wicket. But Ashwin broke the opening stand when he sent back Watson, caught by Sam Curran, for 26.
With spinners bowling from both sides, du Plessis looked to deal in sixes, but it was Ashwin who had the last laugh as he sent back du Plessis (54 off 38 balls) with another carrom ball. Raina (17) too followed suit when he tried to slog sweep, only to miss the ball and see his stumps get rattled.
The veteran off-spinner turned out to be the star with the ball as he picked up all the three wickets that fell to finish with figures of 3/23 from his four overs. While his strikes stemmed the flow of runs, CSK scored 52 runs in the last five overs with Dhoni finishing well along with Rayudu.
However, in the end, Mahendra Singh Dhoni (37* off 23 balls) and Ambati Rayudu (21* off 15 balls) finished with a flourish as CSK made 160/3 in 20 overs.