Kozlov was drafted by the San Jose Sharks in the first round as the sixth overall selection in the 1993 NHL Entry Draft. Kozlov scored his first career goal for the Sharks against Ed Belfour and the Chicago Blackhawks on 21 March 1995 in a 7–3 Sharks loss. He would play a few seasons in San Jose before being traded to the Florida Panthers on 13 November 1997. It would be in Florida, playing with Pavel Bure, that Kozlov would have his best years. His best season was during the 1999–2000 season when he set career highs in assists and points. He played with the Panthers until 1 March 2004, when he was traded to the New Jersey Devils for Christian Berglund and Victor Uchevatov.