Kohli struck two sixes and eight fours off 38 deliveries as the visiting side reached 161-7 in 19.3 overs in reply to Delhi's 160-6 at the Ferozeshah Kotla ground.
For the home side, Australian all-rounder James Hopes top-scored with 54.
Bangalore, which had beaten Kolkata Knight Riders by nine wickets at the Eden Gardens in its last game, registered its third win to reach seven points from seven games with one no-result.
Delhi, which had beaten Kings XI Punjab by 29 runs at the same venue last week, remained on four points from six outings.
For Bangalore, Sri Lankan captain Tillekaratne Dilshan went without scoring but Kohli added 82 for the second wicket with the other opener Chris Gayle, who struck 26 runs from only 14 deliveries.
The two took control, as both new-ball bowlers Irfan Pathan and Ashok Dinda went for 16 runs in an over.
“To be honest I did not plan to bat like that,” Kohli said after being named 'man of the match'. “We lost three in a row and now winning two back to back gives us some momentum.” But Gayle's wicket was followed by three more including that of Kohli, as the Bangalore side slipped from 83-1 to 107-5 in the face of a period of tight bowling led by South African pace bowler Morne Morkel.
Bangalore was left needing 32 off the last four overs but pace bowler Umesh Yadav conceded two boundaries in the penultimate over to concede vital ground for the home side.
Big-hitting left-hander Saurabh Tiwary then smashed 18 before captain Daniel Vettori (18 not out) and Syed Mohammad (13 not out) guided the team through.
Earlier, Delhi was boosted by Australian all-rounder Hopes, who cracked a fluent 54 off 43 balls with seven fours.
Lucky to survive on 43, when Dilshan caught him at backward-point off Abhimanyu Mithun but was asked to return as replays showed it was a no ball, Hopes reached his half-century off 39 balls.
The 32-year-old Hopes, who came in to bat at number three, was out a little later to the same bowler as he tried to heave a delivery - only to nick the ball to wicketkeeper AB de Villiers of South Africa.
The in-form Venugopal Rao (24), wicketkeeper Naman Ojha (16) and all-rounder Irfan Pathan (13 not out) chipped in with useful cameos to help put up a fighting total.
The famed opening pair of Aussie David Warner and Virender Sehwag failed click on a flat pitch.
Warner was bowled by Zaheer Khan for seven, while Sehwag looked good but was run out for 25 in trying to steal a single.
On Wednesday, Pune Warriors takes on Chennai Super Kings at Mumbai, while Kochi Tuskers hosts Deccan Chargers.
