Captain David Warner hit 69 off 38 balls and Ben Cutting a brutal unbeaten 39 off 15 in Hyderabad's 208-7, while England's Chris Jordan took 3-45.
Chris Gayle (74 off 38) and Virat Kohli (54 off 35) added 114 for the first wicket in reply, before Cutting (2-35) removed Gayle.
With 18 runs needed off the final over, Bhuvneshwar Kumar conceded only nine.
Bangalore, who hit 13 sixes in their innings, also lost the 2009 and 2011 finals.
England one-day captain Eoin Morgan, who is part of the Sunrisers squad, did not play in the final in Bangalore.
Sunrisers Hyderabad beat Royal Challengers Bangalore by eight runs to win the Indian Premier League final.