The Sunrisers were made to bat first and hit 172-5 before the Indians were condemned to their fifth consecutive loss with a score of 157-7.
Impressive knocks of 65 and 46 from David Warner and Lokesh Rahul respectively made up the bulk of the Hyderabad runs, while Aaron Finch (16), Darren Sammy (10) and Naman Ojha (10 not out) also helped to beef up their team's score.
Ben Dunk's 20 gave the Indians a solid platform on which to build their run chase following the early dismissals of Anderson and Rohit Sharma, but it was Ambati Rayudu (35) and Kieron Pollard (78) who pushed their team onwards.
However, just 13 runs from the lower order saw Mumbai continue their losing start to the season.
Dale Steyn, Bhuvneshwar Kumar and Irfan Pathan all ended with two wickets each for Hyderabad, while Sammy also took a wicket.