The Meteors reached the target of 122 with two balls to spare at Manuka Oval in Canberra.
Britt claimed 2-29 from four overs with the ball, before adding 38 off 36 deliveries with the bat in a brilliant individual display.
Spirit, who won the toss and elected to bat, posted 121-8 from 20 overs.
Kirsty Lamb blasted three fours and one six in her haul of 27 off 16 balls, while Elyse Villani (24 off 21 deliveries) and Emma Inglis (24 off 27) also proved valuable.
Samantha Bates (3-22 from four overs) and Rhiannon Dick (2-26 from four) were the most damaging with the ball.
In reply, the Meteors seemed in control at 89-2, but lost five wickets for just 25 runs as Spirit roared back into the contest.
Openers Laura Wright (26 off 27) and Asha Smith (21 off 16) ensured a strong start, but were two of only four batswomen to reach double figures alongside Britt and Sara Hungerford (18 off 17).
Villani (3-15 from three) kept Spirit in the contest with support from Emily McIntyre (1-23 from three), Molly Strano (1-19 from four).
But their efforts were in vain as the Meteors scampered home in a thrilling finish.