The competition’s new leaders had got off to a strong start having been put into bat, with Jacob Bethell and Virat Kohli (62) putting on an opening stand of 97.
England batter Bethell hit his maiden IPL half-century, becoming the third-youngest player to pass the landmark at 21 and 192 days.
He was caught by Brevis off Matheesha Pathirana for 55 just short of the halfway mark, with Chennai then putting on a strong bowling display that stalled their opponents’ innings.
Having been 97-1, RCB found themselves 159-5 with 12 balls to go, but Shepherd was to step up in spectacular style.
He blazed four fours and six sixes, with Khaleel Ahmed’s penultimate over going for 33 and Pathirana’s last six deliveries costing 21.
Ayush, 17, almost stole the limelight, hitting 14 boundaries before he was caught in the deep by Krunal Pandya trying to reach a first IPL century with a six.
Ngidi finished with figures of 3-30 after trapping Brevis lbw and having dismissed Sam Curran earlier in the innings, but Dayal took the bowling plaudits as he closed out the victory with some steely death bowling.
The win sees RCB move up two places to first in the IPL table, rising above Mumbai Indians and Gujarat Titans, while Chennai, who are out of contention for the end of season play-offs, remain bottom with two wins from eleven games.