The victor of the Qualifier 1 on Tuesday will remain in Mumbai for the IPL 2018 final to be played on Sunday, whereas the losing team will travel to take on the victor of the contest between Kolkata Knight Riders (KKR) and Rajasthan Royals (RR).
Chennai had fallen to 62-6 but opener Du Plessis guided his side to victory with his 47-ball knock.
CSK skipper Mahendra Singh Dhoni, too, didn't last long. "It's about constantly improving".
"Right from the first game we've used different bowlers in different ways to find out who is the best bowler at the end... They will have to step up in the final", he added.
MS Dhoni managed just nine before being bowled by Rashid Khan, and at the halfway stage Chennai were 50 for four.
Shikhar Dhawan was clean bowled off the first ball of the game by Deepak Chahar, who finished with figures of one wicket for 31 runs.
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Sunrisers' bowling this season have been extraordinary with Rashid and Shakib as well as the pace trio of Bhuvneshwar, Siddarth Kaul and Sandeep Sharma, combining well to defend paltry totals.
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.
Hyderabad captain Kane Williamson, lone foreigner to lead an IPL team this season, hit four crisp boundaries in his 24 but they kept losing wickets at regular intervals.
The 34-year-old Bravo got two important wickets of Shakib Al Hasan and Yusuf Pathan, who was spectacularly caught and bowled by the West Indian allrounder. "Being in the top two gives us that liberty to have one more game", Dhoni says.
"The game did swing our way, and bowlers were outstanding, but we didn't quite execute it like in the past", Williamson said. He has so far scored eight half-centuries in IPL 2018, and he looked in prime form on Tuesday at the Wankhede Stadium in Mumbai before he was dismissed.
KKR and RR will clash in the Eliminator on Wednesday at Kolkata's iconic Eden Gardens.