Leading 3-0 from the opening leg in Iceland, Inter came into the clash at San Siro knowing their qualification to the group stage of the competition was practically assured.
Despite making six changes, Mazzarri's men were ruthless in picking off their opponents to seal a 9-0 aggregate success.
Kovacic, who had failed to score in all of his previous 54 Inter appearances, netted twice in the first half and completed his treble in the second after Pablo Daniel Osvaldo had put the home side 3-0 up.
Mazzarri then introduced Mauro Icardi as a replacement for Kovacic, and the Argentinian added two further goals as Inter completed the rout.
"Kovacic was rewarded for his perseverance," Mazzarri told Sport Mediaset.
"This lad is growing exactly the way we hoped, is becoming consistent and putting his talents to the service of the team.
"The game was pretty much over at 2-0, then it was almost like a friendly. Stjarnan lost intensity and courage. We started well and I was interested to see how we'd react with our heads and our legs.
"We never lowered our guard and that attitude must remain throughout the season. I'm also happy with Osvaldo's fitness levels, Mauro Icardi's attitude and also Hernanes."
Inter will now travel to Torino for the opening game of their Serie A campaign on Sunday.
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