Mitrovic, signed from Newcastle after a loan period last season, struck in the 36th and 38th minutes before James Tarkowski cut the arrears to 3-2 by halftime.
Andre Schurrle scored late on to confirm Burnley's drop into the bottom three.
After finishing seventh last season -- their highest position since 1974 -- Clarets fans will now be concerned that competing in the Europa League is now adversely affecting their domestic performances.
Sean Dyche's team have already played five European games and have another one on Thursday, when they must retrieve a 3-1 deficit at home to Olympiakos Piraeus.
They twice failed to deal with crosses that allowed Mitrovic to beat goalkeeper Joe Hart, one of six players left out for the away leg in Greece.
One of the five survivors, Icelandic international Johann Berg Gudmundsson, had to be substituted in the first 20 minutes
because of injury.
Although Tarkowski's goal before halftime kept them in the contest, Mitrovic could have had a hat-trick. He hit the post before Schurrle knocked in the rebound.
"Every goal is a nice goal, great feeling, but the most important thing is we grab the three points and give the fans a nice afternoon," Mitrovic said.
"We have so many new players, they came from different leagues and some don't speak English but this is the best league, we want to stay with the best and we will work hard to do that."
(Reporting by Steve Tongue, editing by Pritha Sarkar)
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