After Capitals captain Shreyas Iyer opted to put Sunrisers into bat, Warner (66) and Saha (87) established a fine opening partnership worth 107.
Manish Pandey (44 not out) and Kane Williamson (11 not out) helped their team reach 219-2, meaning Delhi needed to achieve the second-highest successful chase in IPL history to win.
Suffice to say they came up short, with the Sunrisers' dismissal of Shikhar Dhawan for a first-ball duck an ominous sign for the Capitals. Marcus Stoinis (five) followed him back soon after having also been caught by Warner.
Shimron Hetmyer (16) and Ajinkya Rahane (26) attempted to repair the early damage but both fell to Rashid Khan, who finished with outstanding figures of 3-7 from his four overs.
Rishabh Pant top-scored for the Capitals with 36 but there rarely looked to be any hope of a remarkable turnaround - they were skittled for 131 with an over to spare.
WARNER AND SAHA COMBINE TO DEVASTATING EFFECT
Warner had got the Sunrisers' charge up and running, reaching 50 from just 25 balls, as they were left sitting pretty at 77-0 at the end of the sixth over, during which the left-hander struck five boundaries.
He eventually fell to Ravichandran Ashwin in the 10th over but Saha – opening in place of Jonny Bairstow, who was left out of the XI – kept going, eventually out 13 runs shy of a century as he tired towards the end of his knock.
RASHID EXCELS WHERE RABADA FAILS
It was a rare off-day for Kagiso Rabada, whose streak of successive IPL matches with at least one wicket ended at 25. That impressive run stretched back to 2017, when he also failed to make a breakthrough in a game against the Sunrisers.
There was no such misery for leg-spinner Rashid, who was in devastating form. His three-wicket haul takes him to 17 for the tournament this year, though that is still six behind Rabada's tally.
Captain Warner (85 not out off 58) and Saha (58no off 45) put on 151 for the first wicket as Sunrisers successfully chased down a victory target of 150 in the final game of the regular season at Sharjah Cricket Stadium, moving them above Kolkata Knight Riders and Royal Challengers Bangalore into third.
They consequently eliminated KKR and will now face RCB in an eliminator on Friday, while the Indians already knew they would be going up against Delhi Capitals in Qualifier 1 on Thursday.
Rohit Sharma (4) was not called up by India for the upcoming tour of Australia but returned to the line-up for the Indians, though he and Quinton de Kock (25) were the victims as Sandeep Sharma (3-34) moved onto 53 wickets from the opening six overs in the IPL – the most by any bowler.
Suryakumar Yadav (36) and Ishan Kishan (33) put on 42 for the third wicket before the former fell to Shahbaz Nadeem (2-19), who dismissed Krunal Pandya in the same over.
Jason Holder (2-25) claimed his 100th Twenty20 wicket when he removed Nathan Coulter-Nile for one with Mumbai 116-7 in the 18th over.
He also accounted for Kieron Pollard after he smashed 41 off 25 balls to push the leaders to a more respectable total, though it was still their lowest when batting first this season.
It proved easy work for Warner and Saha, as their terrific stand got them over the line with 17 balls remaining and kept Sunrisers' campaign going.
POLLARD GOES PLUNDERING
Mumbai rested Hardik Pandya, Jasprit Bumrah and Trent Boult but failed to make things easy for their bowlers after being put in.
Pollard did his best to make the workload a little less daunting with a late flurry, clearing the ropes four times – including three in succession off Thangarasu Natarajan in the penultimate over – as he moved past the 3,000 runs mark in the IPL.
SUNRISERS DON'T GO DOWN
Warner is the leading run-scorer in the IPL and it is easy to see why when he is in such dominating form.
He sent James Pattinson (0-29) to the ropes three times in succession in the fourth over and brought up his 48th half-century in the competition with his only six of a masterful knock.
A wonderful drive between cover and extra cover for his 10th four got the job done for Sunrisers.
