Yashasvi Jaiswal top-scored for RR with 49, while Dhruv Jurel chipped in with 47.
For RCB, Hazlewood (4/33), Krunal Pandya (2/31), Bhuvneshwar Kumar (1/50) and Yash Dayal (1/33) were the wicket takers.
Earlier, Virat Kohli and Devdutt Padikkal struck fifties to help Royal Challengers Bengaluru post 205 for 5.
Kohli (70 off 42 balls) and Padikkal's (50 off 27 balls) second wicket stand of 95 runs was the highlight of the RCB innings after being invited to bat.
Later, Tim David (23) and Jitesh Sharma (19 not out) took charge to take RCB past the 200-run mark.
Popular in Sports
Brief Scores:
Royal Challengers Bengaluru: 205 for 5 in 20 overs (Virat Kohli 70, Devdutt Padikkal 50; Sandeep Sharma 2/45).
Rajasthan Royals: 194 for 9 in 20 overs (Yashasvi Jaiswal 49, Dhruv Jurel 47; Josh Hazlewood (4/33), Krunal Pandya 2/31).