Chennai had fallen to 62-6 but opener Du Plessis guided his side to victory with his 47-ball knock.
Both Chennai and Hyderabad finished the league stage on 18 points, but the Kane Williamson-led side got their neck ahead to the No.1 spot with a superior net run-rate and Tuesday's victor will be guaranteed a place in the final at the same venue while the loser gets a second chance in the second qualifier in Kolkata on May 25.
Needing 140 runs to win, Chennai made heavy weather of what should have been a comfortable chase before crossing the line with five balls to spare.
Khan also accounted for Dwayne Bravo (7), while Sandeep Sharma dismissed Ravindra Jadeja (3) and Deepak Chahar (10) to leave CSK reeling at 92/7.
Wicket-keeper Shreevats Goswami took a brilliant catch to dismiss opener Shane Watson (0) off pacer Bhuvneshwar Kumar.
He hit a six on the first ball of the 20th over to chase down Sunrisers' total of 139 with two wickets left. Read SRH vs CSK (IPL Playoffs' Qualifier 1) Match Scorecard.More news: Warriors list Iguodala as doubtful for Game 4
Siddarth Kaul bowling the 19th over paid for his inexperience in crunch game as 17 runs came off that over as no 10 Shardul Thakur (15 off 5 balls) got three boundaries including a streaky one. Rayudu's dismissal was largely due to poor shot selection as he misjudged a slow yorker only to see the ball crash into the base of middle and off-stump.
Despite constantly losing partners, du Plessis kept his cool and having taken his time to get to the pace of the pitch he launched Brathwaite for 20 runs to reach his half-century.
Du Plessis, on the other hand, started unleashing his shots and hammered few sixes but lack of support from other batters.
The umpire Marais Erasmus signalled it a dead ball as Dhoni collected the delivery, which was surprising as the batsman expected it to be called a no-ball, which would give his team a run and a free hit. Sam Billings is out, Shane Watson is in for Chennai Super Kings. Captain Kane Williamson (24) and Yusuf Pathan (24) were the other notable contributors with the bat.
Earlier, 2016 winners Hyderabad - who topped the league table in this edition - faltered after being put into bat first despite a 29-ball unbeaten 43 from Carlos Brathwaite. Sunrisers will need their middle order, especially Manish Pandey, to take up more responsibility to counter the Super Kings' thrust.
Apart from Jadeja, Dwayne Bravo chipped in with two wickets.