The home fans were expectant following such a bright start to their debut Finals appearance, but they were silenced late in the first period when a sustained period of KalPa attack resulted in Andreas Okany deflecting in Juuso Maenpaa's shot pass.
That was one of only three shots on goal the visitors had in the first 20 minutes, but in the second period, they had 10 to SaiPa's six and made that advantage count midway through, when Jesper Mattila teed up Lasse Lappalainen, who made no mistake with his finish.
Two became three early in the final period, when Kristjan Kombe worked tirelessly behind the net to escape the defence and slot the puck through Kari Piiroinen to score unassisted.
The Lappeenranta side gave themselves hope when Henri Nikkanen powered the puck home two minutes later, only for the three-goal lead to be restored soon after, when Aleksi Klemetti drew Piiroinen and tapped in KalPa's fourth.
The goal trading continued as Antti Kalapudas finished well to take advantage of a powerplay, but pulling the goaltender backfired with a little over two minutes to go, and Patrick Curry found the empty net to complete the 5-2 win.
The series lead will be up for grabs again when the teams meet back in Kuopio for Monday's game three.