Sanju Samson won an important toss and opted to chase.
Virat Kohli got off to another bright start, reaching 30 off 19 in the PowerPlay.
Trent Boult produced an outstanding spell in the PowerPlay, his three overs costing just six runs for the wicket of du Plessis.
Yuzvendra Chahal took the big wicket of Kohli straight after the PowerPlay.
R. Ashwin then produced a masterful middle-overs spell, taking 2 for 19 from his 4 overs.
Rajat Patidar, reprieved early by Dhruv Jurel, was the lone bright spot in RCB's middle-order, scoring a brisk 34.
After conceding 30 from his first 2 overs, Avesh Khan came back well to take 3 for 14 from his final 2 overs.
RR had a mixed day in the field but Rovman Powell was at his best. He took 4 catches.
Yashasvi Jaiswal got Royals' chase off to a breezy start.
Lockie Ferguson dismissed Tom Kohler-Cadmore for RCB's first wicket.