Lundberg and Wiesberger finished tied on 12 under par after the Swede hit a closing 65 and the Austrian 69 after letting a one-shot lead slip with three holes to play.
The pair returned to the par-three 18th for the first extra hole and it was advantage Wiesberger after their respective tee shots, only for Lundberg to hole from 45 feet for an unlikely birdie.
Lundberg had held the halfway lead after rounds of 67 and 68, but a 76 on Saturday left him six shots behind defending Dutch champion Joost Luiten starting the final round.
The 40-year-old hit four birdies in his first five holes and after a bogey on the eighth, carded four more in five holes from the 11th to set the clubhouse target.
"Today just started off really well on the first couple of holes. I made some really nice putts that changed the whole attitude," said Lundberg, whose previous European Tour wins came in Russia in 2005 and 2008.
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"As for the play-off, what can I say? I didn't hit a great shot and Bernd hit a fantastic shot but that's golf."