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Williamson made 81 from 42 balls, including five sixes, and put on 135 for the third wicket with Manish Pandey (62 not out) but the side fell 14 runs shy of chasing down Bangalore's 219 for six.
Needing 20 off the final over, Williamson was caught on the boundary by Black Caps teammate Colin de Grandhomme.
The win keeps the semifinal hopes alive for the Daniel Vettori-coached Royal Challengers, a side that also includes fellow New Zealanders Brendon McCullum and Tim Southee. Sunrisers Hyderabad remain top of the table with one game remaining while the Royal Challengers now sit in fifth.
De Grandhomme scored 40 from 17 balls in the Bangalore innings, including four sixes, while South Africa's AB de Villiers top scored with 69. Sunrisers bowler Basil Thampi produced the worst ever figures in IPL history, going for 70 runs off his four overs.