Australian Marsh hit eight boundaries, including three sixes, in a 34-ball 50 as he shared an opening partnership of 89 with Ryan Rickelton at Headingley.
South African Rickelton made 40 off 22 deliveries for the second-placed SunRisers, while Harry Brook, captain Zak Crawley and Dan Lawrence contributed 23, 34 and 29 respectively in a total of 201-4.
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Opener Matthew Short top-scored for Fire with 43 before being caught by Brook as Carse, who finished with figures of 3-24, claimed his second wicket after removing Phil Salt for 13.
Joe Root added 31 from 22 balls but the visitors fell well short, finishing on 130-6 as their hopes of a top-three finish suffered a major setback.
Meerkat Match Hero Mitchell Marsh said: "In the powerplay it felt like a really good wicket. Once the ball got older, it became tough to bat. The way we adapted with the ball; the guys executed brilliantly."
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