Whenever Sunrisers needed a wicket, Cummins brought himself on, and he usually delivered as the home side were limited to 7-180, the Australian quick taking 3-28 in his four overs.
Though Travis Head failed with the bat the visitors hauled in their target with a perfectly judged run-chase, winning by five wickets with one over to spare.
Ishan Kishan made 70 off 47 balls and Heinrich Klassen 47 off 26, adding 75 in seven overs.
"Really satisfying," said Cummins. "It was a tough wicket to bat on and the way "Klassy" and Ishan chased it down I was really happy. They chose their match-ups and played some ridiculous shots."
"The real stuff starts now," he added.
The defending champions, Josh Hazlewood's Royal Challengers Bengaluru, were already into the last four and Sunrisers' win means Gujarat Titans are also through.
Five teams are in contention for the final place though CSK and Mitch Starc's Delhi Capitals both need an unlikely set of results.
Ricky Ponting's Punjab Kings are currently fourth but have lost their last five matches. Rajasthan Royals have a game in hand on them but are in poor form themselves. Cam Green's Kolkata Knight Riders are also in with a shout.
Asked to bowl in Chepauk on Monday (local time) Cummins began with Nitesh Kumar Reddy and Praful Hinge, but after two overs CSK were 0-30 with Sanju Sansom flying with 27 off 11 balls.
Three balls later it was 1-31 with Sansom caught behind off Cummins without adding to his score.
However, new batter Urvil Patel finished the over with successive sixes and Cummins took himself out of the attack.
As the mid-point approached he returned to bowl the tenth over with CSK 2-89 and Kartik Sharma 32 off 16.
Once again Sunrisers needed a wicket. Once again the Australia captain delivered, dismissing Kartik without the batter adding to his score.
Cummins stayed on and added opposing captain Ruturaj Gaikwad to his victims. The opener, starved of strike went for a boundary-free 15 off 21.
Cummins held back his final over, but felt compelled to bowl it once Dewald Brevis settled in, at 4-132. However, the South African youngster whacked his first two balls for 6, 4. The rest of the over was tighter but there was no fourth wicket.
Brevis went on to a 27-ball 44, the top score as CSK closed on 7-180.
CSK bowled well at the start with Spencer Johnson's first over going for five. Head, given little to put away, attempted a rash smash and was caught and bowled off a skier for a six-ball six by Mukesh Choudary.
When Johnson caught Abhishek Sharma (26) CSK were slight favourites but at 2-75 he dropped Klasasen - a difficult diving chance - on 18 and the South African took full advantage.
A decade on from their one and only IPL triumph Sunrisers have another shot at glory.
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