Kjellberg joined the Canadiens farm club in Fredricton where he scored 10 goals and 37 points in 41 games. His impressive production earned him a seven games with the big team, but Kjellberg was unable to translate his scoring success in the American Hockey League to the NHL game despite directing seven shots on net. Though the Canadiens went on to win the Stanley Cup that spring, Kjellberg was back in the AHL by that point. At season's end, Kjellberg decided to return to Sweden to continue his career.

Over the next six seasons, Kjellberg played professionally in Sweden, but enjoyed his greatest success on the global stage. In 1994 he was part of the Swedish squad that captured the gold medal on Peter Forsberg's dramatic shoot-out goal at the Olympics, then added another gold medal at the 1998 World Championships. When the Nashville Predators joined the NHL as an expansion club for the 1998-99 season, they scoured Europe for players to improve the new team's depth and managed to lure Kjellberg back to the National Hockey League.

Kjellberg managed 11 goals and 31 points during the Predators inaugural season but settled in nicely during his second year with the club. He finished second on the club in points and goals when he fired home 23 markers and added just as many assists. His goal production dipped to 14 the next year, but his point total remained almost the same with 45, which was good for third on the team. Despite his success, his career in Nashville hit a road bump during is fourth season. The development of young rookie Scott Hartnell coupled with a slow start by Kjellberg led to his trade to the Mighty Ducks of Anaheim.

Unfortunately the change of scenery didn't help Kjellberg get his game back on track. In 65 games with the Ducks, he was only able to score seven goals and 15 points.