South Africa's Heinrich Klaasen hit an unbeaten 53 to steer Hyderabad to 197-6, a total their bowlers defended by restricting Delhi to 188-6 at the Feroz Shah Kotla ground.
Marsh returned bowling figures of 4-27 including a maiden over and then hit a 39-ball 63 but the effort was not enough for bottom-placed Delhi, who fell to their sixth loss in eight matches.
Former champions Hyderabad, who won their only title in 2016 under current Delhi skipper David Warner, bounced back from their three losses to move up one spot to eighth.
Warner fell for a two-ball duck but Delhi's chase looked good during a 112-run second-wicket partnership between Phil Salt, who made 59, and Marsh.
Hyderabad hit back with three wickets in three overs through their spinners and once Salt and Marsh departed the chase lost track.
Rising Indian leg-spinner Mayank Markande stood out with figures of 2-20 including getting Salt, a wicketkeeper-batsman from England, caught and bowled.
Batsmen set up victory after left-hand opener Abhishek Sharma smashed 67 off 36 balls after Hyderabad skipper Aiden Markram won the toss and elected to bat first.
Hyderabad seemed to slip after they lost a few wickets but Klaasen, who struck his first IPL half-century, put on a couple of partnerships including a sixth-wicket stand of 53 with Abdul Samad to boost the total.
See the full scorecard on Flashscore.
Titans go top of the table
In the first match of the day, a fine partnership between Vijay Shankar and David Miller steered holders Gujarat Titans to the top of the table with a seven-wicket win over Kolkata Knight Riders.
Chasing 180 for victory, Gujarat were in trouble at 93-3 when Shankar (51) and the left-handed Miller (32) combined to achieve victory with 13 balls to spare at Kolkata's Eden Gardens.
Bowlers Mohammed Shami, Josh Little and Noor Ahmad set up victory after they kept down Kolkata to 179-7 despite a sparkling 81 by Rahmanullah Gurbaz in a match which started 45 minutes late due to rain and wet outfield.
In the chase, skipper Hardik Pandya fell for 26 and Shubman Gill departed on 49 when Miller, who survived a dropped catch on 26, took on the bowlers with a couple of sixes.
Shankar, an Indian all-rounder, soon took over from his South African partner in their unbeaten stand of 87 with a six-hitting blitz as he reached his fifty in 24 balls.
Gujarat, who clinched the IPL title last year in their debut season, moved top of the 10-team table with six wins in eight matches.
