After being put in to bat, the Sunrisers lost opener Aaron Finch inside the first over and, although they managed to get through another 10 overs before losing their next wicket, Dhawan's side never really got going.
Dhawan topscored for Hyderabad with a steady 34-ball 38, while David Warner (32), Lokesh Rahul (20) and Venugopal Rao (16*) also reached double figures to guide the Southern franchise to 133-6 from their 20 overs.
"150 would have been a good total, we were short by 20-30 runs," Dhawan told reporters. "The track was two-paced to start with but once we got our eye in, we were able to play shots. Unfortunately, Warner and I could not carry on [after getting starts] and that put us on the back foot.
"The batsmen need to score more and give the bowlers more to work with."
Despite losing two wickets in the powerplay of their chase, Rajasthan eventually eclipsed the Sunrisers' total with three balls to spare.