The 25-year-old striker signed for the Turf Moor outfit from Middlesbrough earlier this week and made an immediate impression as their pre-season tour got under way with a comprehensive victory.
Jutkiewicz came on at half-time in sweltering conditions in Austria and hit the back of the net in the 61st and 70th minutes.
Burnley went in at the break with a three-goal advantage over their hosts, who ply their trade in the Austrian fifth tier.
Ashley Barnes scored twice in the opening 20 minutes, with Jutkiewicz's fellow new arrival Marvin Sordell replacing the striker and scoring himself nine minutes before the interval.
Burnley boss Sean Dyche made wholesale changes at half-time, but the goals kept on coming through Jutkiewicz and Scott Arfield.
Sordell then got his second of the game in the 83rd minute, before Jason Gilchrist rounded matters off in injury-time.
