2010-11: Ivan Barbashev appeared in nine junior games with Moscow Dynamo and represented Russia in the 2011 U16 Four Nations Tournament. He skated for Dynamo's U17 team in the Russian finals. Barbashev scored 3 goals with 3 assists and had 10 penalty minutes in his stint with the junior team. In two games at the Four Nations Tournament he scored 1 goal and was -1. In 25 regular season games with Moscow's 1995-born team he scored 17 goals with 23 assists and had 46 penalty minutes. In five games during the championship tournament he scored 1 goal with 5 assists and was +3 with 2 penalty minutes.
2011-12: Barbashev moved up to Russia's U22 MHL as a 15-year-old — skating for HC MVD — and played for Russia in three international tournaments. He scored 8 goals with 2 assists and was -5 with 18 penalty minutes in 38 regular season games for HC MVD. In four post-season games he scored 2 goals with 2 assists and was +2 with 25 penalty minutes. In two Four Nation U17 tournaments he had 4 assists in six games. Barbashev was the second-leading scorer for Russia at the 2012 U17 World Hockey Challenge behind Grigori Dikushin. In five games he scored 4 goals with 2 assists and had 4 penalty minutes. He was selected by the QMJHL's Moncton Wildcats with the first overall pick in the 2012 CHL Import Draft and was a second round pick (33rd overall) by Dynamo Moscow in the 2012 KHL Draft.
2012-13: Barbashev joined Moncton after playing for Russia in the 2012 U18 Ivan Hlinka Memorial Tournament and capped the year playing with Russia in the 2013 U18 World Junior Championships. One of two players to skate in all 68 regular season games for the Wildcats, he scored 18 goals with 44 assists and was +9 with 36 penalty minutes. Moncton finished second in the Maritimes Division and lost to Victoriaville in a first round playoff series. Barbashev scored 1 goal with 2 assists and was -3 in five playoff games. In four games at the Ivan Hlinka he scored 1 goal and he scored 3 goals with 6 assists and was +7 with 4 penalty minutes in seven games at the U18 WJC.
2013-14: Barbashev returned to Moncton for his second QMJHL season and was one of 29 players selected to attend Team Russia's camp for the 2013 World Junior Championship in Malmo.