Klapka’s marker was the only goal of the frame, as the Flames held a 13-8 edge in shots - including a Jonathan Huberdeau penalty-shot attempt that Kraken netminder Philipp Grubauer padded away.
Melanson evened the score 2:17 into the second with his first NHL goal, while Grubauer stopped all 17 shots he faced in the middle stanza including a save with the shaft of his stick on a point-blank opportunity from Yegor Sharangovich in the slot.
Shane Wright put the visitors up 2-1 just before the two-minute mark of period three, then Vince Dunn extended the advantage with a snapshot goal just over three minutes later.
Frederick Gaudreau - into the empty net - and Matty Beniers rounded out the scoring with goals 14 seconds apart in the dying embers of the third period.
Grubauer made 41 saves for the visitors, while Dustin Wolf stopped 23 shots for the home side.