RCB bowlers then restricted the defending champions to 160 for 8, despite a fifty by opener Devon Conway (56).
While Maxwell ended with excellent figures of 2 for 22 off his four overs, Harshal was right on the money, snapping three wickets while giving away 35 runs.
Shabaz Ahmed (1/27), Wanindu Hasaranga (1/31) and Josh Hazlewood (1/19) also picked a wicket each.
With the win RCB moved to fourth position on the points table, while CSK were pushed closer to the brink of elimination as they remained at the penultimate spot.
Defending 174, like CSK, RCB bowlers leaked runs in the Powerplay but left-arm spinner Shabaz and Maxwell took a flurry of wickets between the 7th and 10th overs, including that of opener Ruturaj Gaikwad (28).
Conway provided some resistance with a 37-ball 56 but he was sent back to the hut by Maxwell.
With 56 needed off the last five overs, Ravindra Jadeja tried to up the ante but the all-rounder, who has struggled with form all season, hit a slow delivery by Patel over cover straight to Kohli's hands.
RCB's death over specialist then struck again, this time getting rid off the dangerous Ali, who was slowing finding his groove.
When Dhoni walked in CSK needed 52 off 24 balls which soon became 39 from the last two overs. The talismanic leader, however, couldn't get his side over the line.
Earlier, the CSK spinners made full use of Dhoni's decision to bowl first as a fit-again Moeen Ali (2/28) and Maheesh Theekshana (3/27) shared five wickets.
Ali rattled the RCB top-order, getting rid of the opening pair of Kohli and du Plessis, while Theekshana snapped three wickets in the penultimate over.
Dwaine Pretorius (1/42) was the other wicket taker for CSK.
Lomror emerged as the top-scorer with his 27-ball knock, while Dinesh Karthik (26 not out off 17) hit some lusty blows towards the end to help RCB go past the 170-run mark.
Kohli and du Plessis batted with authority as the two veterans pummelled the inexperienced pace duo of Mukesh Choudhary (0/30) and Simarjeet Singh (0/21) in the Powerplay.
The opening pair extracted 57 runs in the first six overs, RCB's highest Powerplay total this season.
With the uncapped talent going for runs aplenty, skipper MS Dhoni introduced spin, which slowed down the proceedings for RCB.
With the pitch offering turn and bounce, Ali, who returned to the side after an ankle injury, broke the dangerous partnership as du Plessis found Jadeja at deep midwicket in the eighth over.
A horrible mix up between Kohli and new-man Glenn Maxwell (3) spelt the end of the big-hitting Australian's short stay in the middle. It was the fourth time that Kohli was involved in a run-out this season.
The CSK bowlers did a tremendous job after an expensive start.
RCB was trying to recover from the twin blows, when Ali bowled a brilliant delivery.
He pitched the ball up outside off, inviting Kohli for the drive. However, the ball turned past the inside edge and clattered the stumps, the former Indian skipper's struggles against Ali's spin continued.
Lomror and Rajat Patidar (21) then shared a quick 44-run stand but Dhoni re-introduced pace in the form of Pretorius and the South African struck in his very first delivery, getting rid of Patidar.
Lomror was dropped on 33 by Jadeja, but the batter couldn't capitalise on the reprieve as he became Theekshana's first victim in the 19th over
The Sri Lankan then went on to dismiss compatriot Wanindu Hasaranga and Shahbaz Ahmed.
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Bengaluru (PTI): BJP leader R Ashoka on Friday claimed that the ongoing power struggle between Chief Minister Siddaramaiah and his deputy D K Shivakumar has created massive confusion in Karnataka.
He remarked that the proverb “when two people fight, the third one benefits” has come true, as AICC President Mallikarjun Kharge is also now making efforts to become Chief Minister.
Speaking to reporters here, Ashoka said the entire government has turned into a house of confusion due to the alleged power tussle.
Kharge’s new statement has only added to the chaos, he said.
Congress chief Mallikarjun Kharge had on Friday said that no date has been fixed yet to discuss possible leadership change in Karnataka.
This came a day after Kharge said there was no change of CM in Karnataka "for now" and that the leadership issue in the state would be resolved soon.
The BJP leader said that Kharge too has aspirations and that he was cheated earlier.
"Perhaps he has received a green signal from the Congress high command to become CM. Everyone is already saying Kharge should become CM. Kharge himself has declared, “I am ready to become CM.” Despite many attempts in the past, he never became CM. Now he is trying to seize the opportunity," the Leader of Opposition in the state assembly said.
According to him, the situation in Congress has perfectly become a case of “two people fighting, third one benefits.”
