Joining forces in the 15th over, Negi (35 from 23 balls) and Jadhav (28 from 22) raised a 54-run stand for the sixth wicket to rescue the visitors, who lost the top five players with just 108 runs on the board.
South African big-hitter AB de Villiers was the other major contributor with a 27-ball 43 for the Bangalore franchise.
Electing to bat, RCB were reduced to 40/2 with skipper Virat Kohli (20) and his new opening partner Mandeep Singh (17) failing to get a decent start for the visitors.
RCB's decision to open with Mandeep backfired as the Punjab batsman after facing 13 balls, failed to get to the pitch of a tossed up delivery from leg-spinner Karn Sharma to be comfortably caught by Hardik Pandya at deep midwicket.
One over later, Kohli, who by now had struck two sixes and faced 14 balls, was undone by a slower one from pacer Mitchell McClenaghan which Mumbai skipper Rohit Sharma caught at short midwicket.
With the team once again in deep trouble, RCB now needed a special from South African star AB de Villiers (43) and Travis Head (12) , who chipped in with a 45-run third wicket stand before left-arm spinner Krunal Pandya ended Head's 15-ball stay.
De Villiers, though raised RCB's hopes of amassing a healthy total by meting out special treatment to Krunal and McClenaghan before the left-arm spinner once again struck to send back the Protea right-hander.
Before being dismissed, the 33-year-old de Villiers faced 27 balls and helped the ball to the boundary on three occasions and thrice over it.
In the very next over, medium pacer Jasprit Bumrah cleaned up Australian Shane Watson (3) to further dent RCB's prospects.
Wicketkeeper-batsman Jadhav and Negi then paired up to add a few valuable runs with the Delhi southpaw taking the aerial route twice off Sri Lankan Lasith Malinga and once off Bumrah before McClenaghan finished off in style by dismissing the duo off successive balls in the final over.
For the hosts, New Zealander McClenaghan was the pick of the bowlers with figures of 3/34 while Krunal chipped in with two wickets.
Brief Scores: Royal Challengers Bangalore 162/8 (AB de Villiers 43, Pawan Negi 35, Kedar Jadhav 28 Mitchell McClenaghan 3/34, Krunal Pandya 2/34) vs Mumbai Indians.
Source - Agency