Chasing 176, Delhi rode a composed innings from Rahul, who marked his birthday with a fluent 57 off 34 before Stubbs applied the finishing touches, ensuring the target was overhauled with a ball to spare.
Put in to bat, Bengaluru opener Phil Salt marked his presence with a half-century to set the tone, but regular wickets in the innings prevented the hosts from accelerating and they finished on 175-8 in oppressive heat.
Bengaluru struck back early in Delhi's chase, with pacer Bhuvneshwar Kumar delivering a telling power-play spell that accounted for three quick wickets, leaving Delhi under pressure at 50-3 inside six overs.
Local favourite Rahul, however, steadied the innings by striking six fours and two sixes in his knock and allowed others to play without pressure despite challenging batting conditions.
"We didn't want to take it too deep, we just wanted to stay positive... and try to put pressure on the bowlers," Rahul said at the post-match presentation.
"There was faith in Stubbs and (David) Miller. They do this on a daily basis for South Africa and franchise cricket."
Stubbs shifted the momentum decisively, seizing control with an unbeaten 60‑run knock to tilt the contest firmly in Delhi's favour, with steady support from captain Axar Patel before he was forced to retire hurt on 26.
Miller (22 off 10) sealed the victory by striking the winning runs, ensuring there were no late alarms.
"I didn't want to take it into the last over. I knew we were going to do it, I didn't know how. But thankfully Miller was at the other end. It's his bread and butter," said Stubbs, who was named player of the match.
The win lifted Delhi to fourth in the standings with six points, while Bengaluru's defeat dented their push at the top of the table, keeping them second on eight points.
The match marked a milestone for the hosts, who became the first team to play 100 IPL matches at a single venue - the Chinnaswamy Stadium. Former Australia captain Allan Border was among the spectators, watching his first-ever IPL match.
Earlier, Bengaluru openers Salt and Virat Kohli began briskly, but the tempo dipped when Lungi Ngidi struck in his opening over to dismiss Kohli for 19 off 13 balls.
Salt, the hosts' top scorer, then took control with a fluent 63 off 38 deliveries, before Tim David added a lively 26 off 17.
"We were 15-20 runs short. The way we started, especially VK bhai (Virat Kohli) and Salt, they gave us a good start. Then we started losing wickets and it put us on the back foot," said Bengaluru captain Rajat Patidar.
Delhi bowlers including Ngidi, Kuldeep Yadav and Patel also did a reasonable job in checking the run flow, with all three picking off two wickets apiece.
Delhi next face Sunrisers Hyderabad on Tuesday, while Bengaluru host Gujarat Titans on Friday.