The AIS mogul skier collected 25.95 points from the judges to finish behind Guilbaut Colas on 26.51, the second consecutive time the Frenchman has edged out the Torino 2006 Olympic champion.
Both skiers were rated equally on turns, earning 14.2 out of a possible 15 points, but Colas was the quickest down the New York State course.
Swede Jesper Bjoernlund took the bronze medal with 25.44 points, while Begg-Smith's AIS team-mate Sam Hall finished in 39th place on 19.85 points.
"Dale skied very well today, excellent technically and in clear control with strong jumps, and he was fast," said Olympic Winter Institute CEO Geoff Lipshut.
"All in all it was a very good day. Dale qualified in third, and had great feel throughout the event."
"A few days' recovery and then the pre-Olympic training camp comes at a good time for Dale, after five World Cup events in just under two weeks."
Begg-Smith leads the World Cup standings on 555 points from Bjoernlund on 412, while Colas, who missed the first two events of the season, is in third position on 370 points.