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With 43 needing off last three overs, Du Plessis, who opened the batting, targetted Carlos Brathwaite’s 18th over as he smashed the bowler for 20 runs which included two sixes and two fours.
Siddarth Kaul bowling the 19th over, paid for his inexperience in crunch game as 17 runs came off that over as Shardul Thakur (15 off 5 balls) got three boundaries including a streaky one.
Then it was Du Plessis, who finished the match having hit five fours and four maximums.
However, SunRisers bowlers deserve credit for making a match of a small total but this time their luck ran out in the end.
Shane Watson (0) edged one off Bhuvneshwar Kumar while Kaul (2/32) castled Suresh Raina (22 off 13 balls) and in-form Ambati Rayudu (0) off successive deliveries.
MS Dhoni failed to read a googly from Rashid Khan while Dwayne Bravo was done in by extra bounce edging one to Shikhar Dhawan at slips.
Du Plessis waged a lone battle as wickets fell in a heap at the other end with Sandeep Sharma getting rid of Ravindra Jadeja and Deepak Chahar.
Earlier, CSK bowlers utilised the lively conditions to the fullest restricting Sunrisers Hyderabad to a modest 139 for 7 after being put into bat.
Effective seam bowling in Powerplay overs swung the game towards two-time champions early on in the innings and they never let it slacken except towards the end when Carlos Brathwaite (43 off 29 balls) laid into Shardul Thakur.
Skipper Kane Williamson (24), Yusuf Pathan (24) and Brathwaite (43 in 29 balls) were the useful contributors for Sunrisers on a track that offered good bounce. CSK pegged back Sunrisers at 47 for 3 with both Williamson and opener Shikhar Dhawan (0) back in the dugout. Dhawan, the second-highest scorer for Sunrisers with 437 runs, was played on off Deepak Chahar of the first delivery of the match.
Williamson, in red-hot form in the season, produced three hits to the fence off the last three balls in the same over, an exquisite extra cover drive being the pick. From the other end, Lungi Ngidi packed off the other opener Shreevats Goswami by accepting a smart return catch, when the batsman tried to pull. At 34 for two Sunrisers, who came into the play-off game after suffering three straight losses, were in trouble and it worsened when they lost their skipper Williamson.
The New Zealand captain feathered a catch down leg-side ball off Shardul Thakur to rival captain Dhoni to leave Hyderabad reeling at 36 for 3 in the fifth over. CSK, attempting to make it to their seventh final in nine seasons that they have been part of IPL, struck another blow just past the Powerplay period.
Shakib Al Hasan also fell to a leg-side catch by Dhoni off Dwayne Bravo (2/25) and it left Sunrisers at a spot at 50 for 4. After 10 overs, the score was a modest 64 for 4.
Ravindra Jadeja (1 for 13) sent back Manish Pandey cheaply as half the Sunrisers was back in the pavilion with only 69 on the board. Then Bravo pulled off a stunner, pouching on to a low full-blooded drive from Pathan.
A late flurry from West Indian Brathwaite, who slammed Thakur for four huge sixes and a four in death overs, boosted the total significantly.
(With PTI inputs)