Having been a set a target of 165 by Dolphins to win the Group A game in Mohali, the Scorchers appeared set to for defeat after strong death bowling from Robbie Frylinck (1-32) and Kyle Abbott (1-25) had curtailed the strong start to their response.
But with 12 needed off the final two deliveries, Marsh (40) produced a stunning display of hitting by smashing a pair of full tosses from Frylinck over the boundary to secure a six-wicket success for the Big Bash League champions.
The Dolphins started their innings badly, losing two wickets inside the first over but Khaya Zondo (63 not out) led their recovery with a 50-run innings that included seven fours.
Zondo was supported by Keshav Maharaj (29) and Daryn Smit (21) as they pushed towards three figures and a seventh-wicket partnership of 44 with Andile Phehlukwayo (14) brought the total close to 150.
Frylinck added 15 from six balls at the end of the innings to give the Dolphins a respectable 164-7 from their 20 overs, with Joel Paris (2-21) and Jason Behrendorff (3-46) doing most of the damage with the ball.
After losing captain Adam Voges (7) in the fourth over the Scorchers were kept in the game by Craig Simmons (48) and Sam Whiteman (45).
The pair added 55 for the second wicket and by the time both men were out the Australian side were 118-3 in the 16th over.
Marsh's innings had not seen many fireworks until the final over with just three boundaries, however with the help of Ashton Agar they needed 16 off six balls to win.
A single off the first ball saw Agar on strike and he promptly gifted a catch to Alexander at short leg on the second delivery.
Ashton Turner took three off the next two balls to set up Marsh for his sensational finish.
The fifth ball was sent miles into the stands over midwicket to keep the dream alive and the final delivery, an attempted yorker, was dispatched over the bowlers's head to cue huge celebrations from the Scorchers' bench.
Victory will give Perth confidence ahead of their clash with Kolkata Knight Riders, who started their campaign with a victory.