Despite the win, RCB finished at the bottom of the table with just seven points from 14 games, while Daredevils were placed sixth with 12 points from as many matches.
The home side needed 13 in the final over after Mohammed Shami slammed debutant pacer Avesh Khan for a six and a four. But Daredevils could manage only three and were bowled out for 151.
Earlier, skipper Virat Kohli led from the front with a 45-ball 58 to guide RCB to 161 for six.
Negi was RCB's most successful bowler, finishing with figures of 3/10 in two overs.
Daredevils lost Sanju Samson early, off the innings' second ball, while Karun Nair gave away his wicket after scoring 26 with the help of five boundaries to leave Daredevils at 41 for two in the sixth over.
Samson's wicket was young pacer Avesh's first. It was surprising that RCB opted for the 20-year-old in only their last game of what has been a forgettable season.
While Iyer chipped Travis Head on the one side for a four, Pant went inside-out and cleared long-off to tally 16 runs off the spinner's over.
At the end of the 10th over, Daredevils were 76 or two, needing 86 runs off 60 balls.
In the 12th over, seamer Harshal Patel dealt RCB a double blow when he sent back Iyer and Marlon Samuels off successive deleveries, to give RCB the upper hand and leave Daredevils in trouble at 87 for four.
With 49 needed in four overs, Pant sent Harshal Patel soaring over the square leg boundary for a huge six, but the bowler fought back strongly when he bowled the left-hander to swing the game in his side's favour.
Shami dispatched the last two balls for boundaries to bring the equation down to 35 off 18 balls.
Amit Mishra beautifully played Negi over his head for a six, but was stumped two balls later, as Daredevils required 27 in two overs, which in the end proved to be a tall ask.
The big-hitting West Indian opener then sliced Shami for a six over the third man region, and if that was not enough, he moved across a little to loft the seamer's slower delivery over deep midwicket for another maximum.
Even as Gayle threatened, Daredevils got the first breakthrough when Cummins rattled Vishnu Vinod's stumps after getting a bottom edge, the bowling change working for the home team.
Gayle drove Shami for a boundary to long-on in the first ball of the sixth over, but the seamer came back well in the following deliveries to keep the left-handed batsman in check.
Cummins bowled a tight seventh over before Kohli broke the shackles by hitting Mishra for a boundary.
It was characteristic Kohli as the Indian captain timed one superbly to send Anderson over long-off for a maximum, bringing the side's fifty in the first ball of the ninth over on a slow pitch.
Languishing at the bottom of the table, RCB were 66 for one at the halfway stage.
The fifty-run partnership for the second wicket was achieved in 41 balls, thanks to a productive 12th over that fetched 19 runs.
However, Gayle fell two runs short of a half century, as Shahbaz Nadeem had him caught at midwicket for a rather soft dismissal.
Meanwhile, Kohli brought up his fifty -- his fourth of the season -- off 39 balls, but got out soon after he clobbered Zaheer over deep midwicket for his third six in what could be the former India seamer's final IPL game.
Disclaimer: No Business Standard Journalist was involved in creation of this content
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