Born in Russia, Jaskin was eight months old when his family relocated to the Czech Republic after his hockey-playing father, Alexej, joined a club there. Raised in the Czech Republic, the younger Jaskin would represent his adopted homeland internationally.
Jaskin started his career in HC Vsetin's youth programs and eventually transferred to Slavia Prague, making his debut for the club in the Czech Extraliga in 2010-11 at age 17. He had three goals and 10 points in 33 games, making him the second highest scorer in the league among junior-aged players. The St. Louis Blues selected Jaskin in the second round (No. 41) of the 2011 NHL Draft.