When Kohli sliced Cottrell to Ravi Bishnoi after Devdutt Padikkal sent the West Indies left-armer to the same man at mid-on, RCB were left with a mountain to climb.
Bishnoi (3-32) also claimed the scalps of Aaron Finch (20), Umesh Yadav (0) and Sundar to cap an impressive display, but the pick of the bowlers was Murugan Ashwin.
His leg spin accounted for De Villiers and put RCB out of their misery, with Navdeep Saini (6) and Yuzvendra Chahal (1) the final wickets to fall as Ashwin finished with figures of 3-21 from his three overs.
Royal Challengers Bangalore: Devdutt Padikkal, Aaron Finch, Virat Kohli (c), AB de Villiers, Shivam Dube, Josh Philippe (wk), Washington Sundar, Navdeep Saini, Umesh Yadav, Dale Steyn, Yuzvendra Chahal.