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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Lucknow (PTI): The Uttar Pradesh Congress on Wednesday staged a statewide protest demanding a fair and transparent inquiry into the FIR lodged against Swami Avimukteshwaranand Saraswati and those who filed the complaint against him.
In a statement issued here, the party said memorandums addressed to Prime Minister Narendra Modi were submitted through district magistrates in all districts of the state.
Uttar Pradesh Congress spokesperson Manish Hindvi told PTI that the memorandums were handed over through the district administration in all 75 districts.
In the memorandum, the party alleged that Saraswati and his disciples were "unnecessarily harassed and humiliated" by police on the occasion of Amavasya and were prevented from taking a ritual bath (at the Magh Mela). It further alleged that some disciples were manhandled and taken to a police station.
The memorandum also claimed that an FIR was later registered against Saraswati, his disciple Swami Mukundanand Brahmachari and several unidentified persons in a sexual harassment case. It termed the case a "conspiracy" aimed at tarnishing the seer's reputation.
Citing Articles 25 and 26 of the Constitution, the memorandum stated that these provisions guarantee religious freedom and the right of religious denominations to manage their own affairs.
It described the position of shankaracharya held by Saraswati as "one of the highest spiritual posts in Sanatan tradition" and alleged that the entire episode appeared to have been "orchestrated in a planned manner".
"We request that the background of the persons who got the FIR registered be investigated in a transparent manner by a retired high court judge and strict action be taken against them," the memorandum said.
It also sought a "fair and transparent probe" into the allegations levelled against Saraswati so that the truth could be established.
Earlier, Uttar Pradesh Congress president Ajay Rai had told reporters in Varanasi after meeting Saraswati that the party stood firmly with him.
The Congress said it would continue to press for an impartial inquiry into the entire episode.
On February 21, an FIR was lodged in Prayagraj against Saraswati and his disciple Mukundanand Brahmachari on charges of sexually abusing two persons, including a minor, over the past year at a gurukul and religious congregations, including the recently concluded Magh Mela.
Days after he was booked, Saraswati had said on Monday that he would not oppose his arrest and asserted that the "fabricated story" would be exposed sooner or later.
At a press conference on Wednesday, Saraswati alleged that criminals rule in Uttar Pradesh, level allegations and influence investigations, as he denied having any contact with the two persons for whose alleged sexual abuse he has been booked.
