Both teams received a point each but that wasn't enough for defending champions Kolkata, who were eliminated from top-four contention with only one game remaining.
Bengaluru climbed atop the table and need only one more point to secure a place in the knockouts.
The IPL was suspended for a week due to the military showdown between India and Pakistan. The Pakistan Super League also resumed on Saturday.
The IPL final, originally scheduled for May 25, has been moved to June 3.
Saturday's game was also supposed to mark an emotional return to the playing arena for India great Virat Kohli, who announced his Test retirement last Monday.
Fans arrived in great numbers wearing a white jersey to honour the ace batter, who played 123 Tests for India and finished just short of 10,000 Test runs.
Kohli has 505 runs in this IPL, the fourth-highest tally. However, he never stood a chance of batting at home thanks to incessant rain in Bengaluru since Friday evening.
The Royal Challengers have two remaining round-robin games against Sunrisers Hyderabad on May 23 and Lucknow Super Giants on May 27, while Kolkata's last game is against already-eliminated Hyderabad on May 25.
Sunday's double-header features Rajasthan Royals hosting Ricky Ponting-coached Punjab Kings in Jaipur, and Delhi Capitals, without their key quick Mitchell Starc who's decided not to return to India, taking on Gujarat Titans in New Delhi.
