Bozek turned pro with the Kings after three high scoring season with Northern Michigan University. An All-American in 1981, Bozek skipped his last year of University in order to jump directly to the NHL.
Bozek made quite the impression in his first season with the Los Angeles Kings, who made him the 52nd overall draft pick in the 1980 Entry Draft. At the time the famous Triple Crown line was hurting as left winger Charlie Simmer missed much of the season with a broken leg. Coach Parker MacDonald placed the speedy but small Bozek on the left side of Marcel Dionne and Dave Taylor.
Steve exploded for 27 goals in his first 34 games, only bettered by a young Wayne Gretzky. Bozek's pace would have seen the rookie score 64 goals had he played the entire season with Dionne and Taylor.
"Subconsciously I was telling myself that this has got to end (the goals) but while it lasted it definitely was fun." says Bozek.
The goals did come to a screeching end once Charlie Simmer returned to the lineup and assumed his place on Dionne's and Taylor's left side. Bozek would only score 6 more goals the entire season. Steve ended his first NHL season with 33 goals establishing a new club record for rookies (since broken) breaking the old mark set by Mike Byers in 1970-71 (27).
Simmer's return coupled with another high scoring rookie called up half way through the season, Bernie Nicholls, forced Bozek to change his game. Because of his great speed and agility, the Kings, who could score but desperately needed players who were willing to sacrifice their offensive totals to help prevent goals against, asked Bozek to become a penalty killer and checker.
"During my first year in L.A. we went through a coaching change mid-way through with Don Perry replacing Parker. Despite the fact that I was scoring well the team had to tighten up defensively so my role became more expanded and I started killing penalties. As it turned out penalty killing became my forte throughout my career."
Bozek struggled through a injury plagued sophomore-jinx filled second season only scoring 13 goals and 13 assists in 53 games. It proved to be the final season in Los Angeles for Bozek, who in the summer would be traded to Calgary for Carl Mokasak and Kevin LaVallee.
Calgary at the time was building an offensive powerhouse to compete with their Albertan rivals from Edmonton. Bozek, however, wasn't brought in specifically for his offense, but rather because of his speed and checking abilities which would become unheralded in many Battles of Alberta. He enjoyed almost 6 full seasons in Calgary, and even managed his only other 20+ goal season in 1985-86 when he scored 21 goals in just 64 contests.
The Flames included Bozek in a late season trade package in 1988 which is often credited with giving the Flames their first Stanley Cup Championship. Bozek, who suffered a terrible injury plagued season, was a throw-in (though Flames GM Cliff Fletcher was very reluctant to part with him) with a hot Flames prospect named Brett Hull. The pair went to St. Louis in exchange for Rob Ramage and Rick Wamsley, both of whom would prove to be big parts of the 1989 Stanley Cup squad.
The Flames and Blues became favorite trading partners, as Fletcher liked to pick St. Louis Blues GM Ron Caron's pocket. A series of trades involving the two always seemed to end up in Calgary's favor. In the summer of 1988, the Flames went after Blues stalwarts Doug Gilmour and Mark Hunter and insisted Bozek be returned to the Flames (along with defensive prospect Michael Dark). The Flames gave the Blues Mike Bullard, Craig Coxe and Tim Corkey in return. Fletcher really appreciated Bozek's speed, intelligence, and work ethic.
However because of the Flames depth, they were forced to either trade Bozek or lose him in the pre-season waiver draft. The Flames would not be able to protect the speedster and ended up trading him along with long time (though injury prone) blueline star Paul Reinhart to Vancouver for a draft pick.
In Vancouver, Bozek mainly became a 4th line center, often centering the Canucks "Club Chaos" line with Rich Sutter and Stan Smyl. The trio was relentless in their persuit of the puck, and though all were small players (Bozek at 5'11" 180lbs was the biggest) they played a fearless style and became instant favorites in Vancouver. Local broadcaster Tom Larscheid often referred to the three as Hack, Smack and Whack, in no particular order!
Bozek, who was playing in his home province, enjoyed his three seasons with the Canucks, but in 1991-92 he left when he signed a 1 year contract as a free agent with the expansion San Jose Sharks. Bozek had a tough season in San Jose, scoring only 8 times in 58 games. As it turned out, it was Bozek's last season in the NHL.
Bozek continued to play hockey in Italy until 1993 when he retired and returned to school. This time Bozek went to Harvard and became a real estate entrepreneur.