The native of Moscow played several years for both the Soviet Wings and the Central Red Army. He helped the Soviet squad win consecutive gold medals at the World Junior Championships in 1983 and 1984. Nemchinov became a regular on the USSR national team and took part in Rendezvous and the Canada Cup in 1987, and the World Championships in 1989, 1990, and 1991. He also toured the NHL with the Red Army in 1989 and 1991 and the Wings in 1990.

The Rangers chose him 244th overall in 1990 on the off chance they would be able to get him released from his commitments back home. As the Soviet Union began to disintegrate in the early '90s, Nemchinov became available. He scored 30 goals and helped the Rangers finish with the most points in the NHL during the 1991-92 season. Nemchinov became a multi-purpose forward for the club and helped them break a 54-year Stanley Cup drought in 1994.

By the mid-'90s Nemchinov's role was more defensive and the Rangers' fortunes declined. The skilled centre played for Russia at the 1996 World Cup where he scored three points in five games. He was traded to Vancouver in March 1997 but the team missed the playoffs and he signed as a free agent with New York Islanders in the off-season. Nemchinov was a solid two-way performer for the woeful club and played for Russia at the 1998 Nagano Olympics and the World Championships a few weeks later.

Late in the 1998-99 season, the veteran was traded from the struggling Islanders to the powerful New Jersey Devils. His skill and checking ability were highly valued by Jersey and he scored three goals and won many face-offs when the team captured the Stanley Cup in the spring of 2000. In 2000-01 he continued to play a solid role as the club reached the finals for the second straight year but lost the seventh and deciding game to the Colorado Avalanche.