But as the Iron Curtain lifted and Russian players started fleeing to the National Hockey League, Kvartalnov attracted little interest. But he was determined to come to North America. He signed with the San Diego Gulls of the IHL. The Californian sunshine compared to the deep Russian winter was an immediate win for him.
The move played out better than even Kvartalnov could have hoped. Playing alongside Len Hachborn and a young Ray Whitney, Dmitri exploded for 60 goals and 118 points in 77 games. He was the face of the IHL, a league which was becoming more and more respected in the 1990s.
This was Dmitri's ticket to the National Hockey League. The Boston Bruins, one of the last teams to get in on the Soviet exodus, selected Kvartalnov 16th overall in the weak 1992 NHL Entry Draft.
At first the fit seemed better than perfect. Playing alongside Adam Oates and Joey Juneau, Kvartalnov exploded out of the gates in record fashion. The 26 year old set a rookie record (since broken) by scoring at least one point in each of his first 14 NHL games. Kvartalnov tallied 12 goals and 10 assists in his impressive debut. He would slow down, but still finished his rookie season with 30 goals and 42 assists for 72 points in 73 games.
But all was not well on the inside. Surprisingly the Bruins left him unprotected in the expansion draft, and despite his rookie campaign. Perhaps more surprisingly, he went unclaimed.
Upon his return to Boston for year two he immediately landed in coach Brian Sutter's dog house. Kvartalnov was a soft, individualistic player who shied away from the physical game - pretty much the exact type of one-dimensional player Sutter could not stand. After half a season Kvartalnov was waived to the minor leagues, never to return to the NHL again.
Kvartalnov left North America at the end of that 1993-94 season. He found big money in Europe, which is where he played for another 13 seasons before retiring in 2007. He starred in Switzerland, Austria, Germany, Finland, and of course back home in Russia with several teams, most notably Kazan Ak-Bars.