2007-08: Dmitry Korobov skated in 47 games for Gomel in the Belarus League as an 18-year old. He scored 3 goals with 7 assists and had 36 penalty minutes. He represented Belarus in the 2007 World Junior A U19 Challenge and the 2008 U20 Division 1A World Junior Championship. He had 2 assists in four games at the WJAC and was plus-four with 1 goal and 1 assist in five games at the WJC-D1A tournament.
2008-09: Korobov made his KHL debut – skating for Dinamo Minsk – and played for Shinnik Bobruysk in the Belarus league. He represented Belarus in the 2009 IIHF World Championship and U20 Division 1A World Junior Championship tournaments. Dinamo Minsk had the KHL's third worst record and Korobov was minus-five with 2 assists and 9 penalty minutes in 12 games. He scored 1 goal with 10 assists and 32 penalty minutes in 24 games for Shinnik. Korobov saw limited ice time in three games for the men's team at the WC, averaging 5:16 minutes of ice time as Belarus finished eighth. He had 1 goal and 2 assists and was plus-five in five games for the U20 team, which narrowly missed elevation to the top group after losing 2-1 to Switzerland.
2009-10: Korobov missed much of the season due to injury. He appeared in four games for Shakhter Soligorsk in the Belarus League.
2010-11: Korobov joined Dinamo Minsk on a full-time basis, skating in 31 games, and represented Belarus in the IIHF World Championship for the second time. He scored 1 goal with 6 assists and was minus-three with 24 penalty minutes for Dinamo. After finishing fourth in the Tarasov Division, Dinamo took first-place Yaroslavl to seven games in a first round series. Koroboov appeared in six playoff games and was plus-one with 1 goal, 2 assists and 10 penalty minutes. He was the leading scorer amongst defensemen for Belarus, which finished 14th at the WC. He scored 2 goals with 2 assists and was plus-four in seven games.
20-11-12: Korobov played in 39 games for Dinamo Minsk in his second full season in the KHL and led Belarus in ice time at the 2012 IIHF World Championship. Skating alongside former NHL player Lukas Krajicek for Dinamo he was plus-12 with 1 goal and 10 assists and had 16 penalty minutes. Dinamo finished fourth in the Tarasov Division. Korobov played in 3 of 4 playoff games against Dynamo Moscow and was minus-three with no points and 2 penalty minutes. In seven games at the WC he had 3 assists and was minus-three with 4 penalty minutes. Belarus finished 14th in the 16-team tournament. Korobov signed a two-year contract with Tampa Bay in August, 2012.
2012-13: Korobov played for Tampa Bay's AHL affiliate in Syracuse in his first pro season in North America. He scored 3 goals with 19 assists in 65 games and was +20 with 34 penalty minutes. The Crunch finished first in the East Division and reached the AHL Finals against Calder Cup champion Grand Rapids. Korobov scored 1 goal with 1 assist and was -2 with 9 penalty minutes in 17 playoff games.