Both the batters took time to get going and had a few lucky escapes, including when Warner was dropped on 19 by Jos Buttler and he survived another reprieve when the ball hit the stumps but the bails did not come off. The two chances were in Yuzvendra Chahal's first over.
Marsh, who was hospitalised last month after testing positive for Covid-19 as Delhi suffered a virus outbreak in their squad, reached his maiden IPL fifty with a six off Chahal.
Marsh, who hit five fours and seven sixes in his 62-ball knock, finally fell to Chahal and missed out on a hundred.
The swashbuckling Warner was slow by his standards but anchored the chase to perfection and hit the winning runs.
In the Delhi innings, the in-form Buttler -- the tournament's leading run-scorer with 625 runs including three centuries -- fell early for seven before Ashwin rebuilt the innings.
Ashwin, promoted to number three in the batting order, put on key partnerships including a 52-run third-wicket stand with Devdutt Padikkal, who made 48, but departed soon after getting his first IPL half-century.
Padikkal kept up the attack as he finished with six fours and two sixes in his 30-ball knock as Royals managed a competitive total despite some disciplined bowling.
Left-arm quick Chetan Sakariya stood out with figures of 2-23.