Rayudu's knock came off 54 balls and it featured six fours and four sixes. Pollard, who hit five fours and three sixes, raised his half century when he hit countryman Chris Gayle for a winning boundary in the last over.
Needing 14 to win, Pollard flicked off-spinner Gayle over the midwicket fence for a six before hitting two successive boundaries.
Earlier, Mayank Agarwal hit six boundaries and four sixes in scoring an unbeaten 64 off only 30 balls to lift the Royals total, hitting seamer Munaf Patel for 24 runs in the last over of the innings.
Mumbai Indians now have 18 points from 14 matches. They share top spot with Delhi Daredevils, who are on the same points, but have a game in hand on Mumbai.
Royal Challengers are in fifth place in the nine-team league with 15 points after winning seven of its 14 matches.
AP
