Renegades spinners Brad Hogg and Narine each took wickets in quick succession to leave the game in the balance before Broad saved the win.
George Bailey made 59, while Narine took 3-27 for the Renegades.
The Renegades earlier looked well-placed after posting a BBL record 4-222, helped by half-centuries from Aaron Finch and Tom Cooper.
It was another impressive performance from Finch, who responded to being dropped from Australia's one-day squad with 57 during the Renegades' loss to the Sydney Sixers on Monday.
Fellow veteran Cameron White chipped in with an entertaining 34 with the Hurricanes bowlers offering little resistance.
But they were soon overshadowed by McDermott's heroics. The son of former Australian Test quick Craig McDermott was playing just his second game of the BBL season.