Although the RCB bowlers managed to keep the Pune batsmen under a tight leash restricting them to 157 for three but another shoddy batting performance saw them finish at a dismal 96 for 9 after 20 overs.
While Pune kept themselves in contention with their fifth win in nine games, RCB are practically out of the tournament with their seventh defeat in 10 games and only five points in their kitty.
Pune bowlers rose to the occasion defending a sub-par total with a controlled bowling performance.
Imran Tahir was the pick of the bowlers with figures of 3/18 from four overs, while Lockie Ferguson claimed 2/7 in four overs to claim the Man of the match award.
Jaydev Unadkat, Daniel Christian and Washington Sundar took one wicket each.
This is one of the rarest occasions when a team has been bundled out below 100 runs twice in the Indian Premier League.
RCB were shot out for 49 by KKR in an earlier game in the tournament.
Chasing a modest 158, Bangalore just kept losing wickets despite skipper Kohli holding his ground at one end.
Apart from Kohli, who came up with an uncharacteristic 48-ball-55, studded with four boundaries and a six, none of the other RCB batsmen could reach the double-digit mark.
Though not in his usual aggressive self, Kohli single- handedly tried his best to bail the team out of trouble but just did not find an able partner at the other end.
The highest partnership of just 21 runs tells the dismal saga of a vaunted batting line-up.
The visitors were gasping at 48 for five in the 10th over and there was no time when RCB showed any signs of revival.
Kohli only reduced the margin of defeat with a gritty knock.
Earlier, RCB bowlers dominated the proceedings from the word go after inviting the Pune outfit to take first strike, with left-arm spinner Pawan Negi returning with impressive figures of 1/18 from his 4 overs.
Stuart Binny (1/17) and Samuel Badree (1/31) chipped in with wicket apiece.
Despite losing just three wickets, Pune batsmen could not upped the ante even though the pitch was reasonably batting- friendly.
Skipper Steven Smith top scored with 32-ball-45 that included five boundaries and a six, while Manoj Tiwary (44 no, 35 balls) and opener Rahul Tripathi (37) also made important contributions.
However, Pune could bring up 100 runs only in the 14th over.
Even Mahendra Singh Dhoni, who hit a huge six and a boundary during his unbeaten 21 off 17 failed to increase the tempo in the final overs. Dhoni and Tiwary could put up only 46 runs in the last five overs.
Despite three 40-plus partnerships, Pune batsmen simply failed to get going, even as the bowlers maintained a consistent line and length to tightened the noose.
Having lost Ajinkya Rahane (6) early, opener Tripathi and Smith stitched 40 runs for the second wicket, before the skipper joined hands with Tiwary for a 50-run stand for the third wicket.
Tiwary then couple with Dhoni for an unbroken fourth- wicket partnership of 49.