England captain Heather Knight's 52 helped Storm chase down Loughborough's total with three balls remaining.
Ellyse Perry (64 not out) had scored more than half of Lightning's runs as they posted a below par 124-7.
Storm reached 124-3 by their 19th over but then lost two quick wickets, before Georgia Hennessy hit the winning runs.
Table-topping Southern Vipers await in the final, which begins at 15:00 BST on Sunday.
Victory saw Storm avenge their five-run loss to Lightning in the group stages, their only loss in the tournament so far.
West Indies captain Stafanie Taylor toppled Lightning's top order, bowling Dane van Niekerk and trapping Georgia Elwiss lbw, before Sophie Devine and Perry's 58-run third-wicket stand steadied the ship.
Perry reached her first half-century of the Super League and then the Australia all-rounder and Thea Brookes together smashed 20 off the final over to trudge past 120.
With Lightning needing a quick breakthrough, South Africa all-rounder Van Niekerk made up for a poor show with the bat as she bowled Rachel Priest.
However, tournament top-scorer Taylor and Knight shared 57 off 50 balls to put their side in control and kept alive Storm's hopes of lifting the inaugural Super League trophy.
Western Storm set up a Super League final against Southern Vipers as they eased to a five-wicket semi-final victory over Loughborough Lightning.