Mohammed Siraj ended with 3 for 27 that dented the Royals top order early.
Faf du Plessis and Deepak Chahar set up Chennai Super Kings 18-run win over Kolkata Knight Riders.
Sunrisers finally seal a chase as bowlers, Bairstow shine with unbeaten half-century.
Dhawan continues to be the Capitals premier batsman, guiding the tricky chase with a 42-ball 45.
With Royals slipping from 87 for 2 to 95 for 7, they were beaten long before the match ended
Punjab Kings suffered a six-wicket defeat in Mumbai, their 196 target proving insufficient.