Competing for the first time since January 2010, the Torino Olympics gold medallist earned 24.13 points from the judges to finish just 0.28 points off the gold medal and 0.06 out of bronze.
AIS/OWI team-mate Ramone Cooper finished in ninth position on 23.48 points, claiming the second top ten result of his career.
Sweden's Jesper Bjornlund collected the gold medal, and his first podium, with 24.42 points.
US skier Bryon Wilson also made his first podium, taking silver on 24.34 points, while defending World Cup champion Alexandre Bilodeau of Canada rounded out the top three on 24.19 points.
Begg-Smith showed he was back near his best by leading the points score for turns, but Australia's third competitor, Sam Hall, had a disappointing day, finishing in 55th place on 12.39 points.
A second World Cup will be held tonight at the Finnish resort.