Veilleux, from Cap Rouge, Que., won the 121-kilometre trek around the Swiss Alpine village of Champery in three hours 17 minutes 35 seconds.
"It comes from a lot of sacrifices and huge amounts of work," said Veilleux, racing in the Dauphine for the first time.
Meersman came 1:56 back.
Veilleux formed part of an early breakaway along with France's Jean-Marc Bideau and Thomas Damuseau, and Spaniard Ricardo Garcia Ambroa. The peloton let them go and they built up a lead of more than 10 minutes at the halfway point of the stage.
The 25-year-old Veilleux made his decisive break about 50 km from the end when he attacked up le Col du Corbier.
"Normally I'm not a climber, but I know that I can climb decently," Veilleux said. "It was worth trying."
Race contenders Alberto Contador and Alejandro Valverde rolled in with the main pack, one second behind Meersman and 1:57 adrift of Veilleux.
Monday's second stage is a 191-km ride from Chatel to Oyonnax.