The former Leicester City player took an Ivan Perisic pass in his stride and beat goalkeeper Ilie Cebanu with a crisp low shot from 13 metres in the ninth minute.
The 24-year old Kramaric, who made no impact during his one-year spell at Leicester before he was loaned to German Bundesliga side Hoffenheim in January and made the move permanent this week, was substituted at halftime.
"I am fairly satisfied given that we played after a lot of hard-working practice sessions but our finishing left something to be desired," Croatia coach Ante Cacic told the country's national HRT television.
Cacic fielded a makeshift formation missing most of the team's key players and gave an array of youngsters a chance to impress as he made sweeping changes at the interval.
The Croatians produced a drab performance for much of a hot and humid evening in Koprivnica, with the final ball letting them down in the absence of Real Madrid playmaker Luka Modric and Barcelona midfielder Ivan Rakitic.
Modric is set to feature in Saturday's Champions League final against city rivals Atletico in Milan while Rakitic and several other stalwarts were rested at the start of Croatia's build-up for next month's tournament in France.
The home side missed several gilt-edged chances as defender Sime Vrsaljko, forward Duje Cop and 19-year old midfielder Ante Coric hit the woodwork while Nikola Kalinic twice headed inches wide from close range.
Croatia will play San Marino in a friendly on June 4 before heading to the 32-nation tournament, where they were drawn in Group D alongside holders Spain, the Czech Republic and Turkey.
(Writing by Zoran Milosavljevic in Belgrade; Editing by Ed Osmond)
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