The 28-year-old Czech latched onto a Laurent Mueller assist to end the match -- and the series -- after 8:24 of overtime.
Thornton, a 25-year-old center who can normally be found captaining the Boston Bruins, finished the season as the playoffs' joint top points scorer with four goals and 20 assists.
The Canadian international's 20th assist came on Thursday when he and Nash combined to set up Swiss international Michel Riesen for the night's fourth goal.
Nash, a 20-year-old winger for the Columbus Blue Jackets, scored nine goals for Davos during the playoffs with Florida Panthers winger Hagman getting 10.
"It feels great, I've never really won anything in my life," Nash told Swiss television on Thursday.
"Hopefully the Stanley Cup will be next, but now I just want to celebrate the Swiss championship."
One of Switzerland's most traditional clubs, Davos have now won 27 national league titles.
All but six of those wins date back to the first half of the last century, however, when Davos were the dominant force in Swiss hockey.
Thursday's victory was the club's second league title in the past 20 years.
Davos coach Arno Del Curto took an early gamble that the NHL lockout would not be resolved in time to rescue the North American season, provisionally signing up Thornton, Nash and Hagman during the summer.
If the NHL dispute, predominantly over players' salaries, had been settled, the club from east Switzerland would have instantly been deprived of their three top players.
Former Atlanta Thrashers team mates Randy Robitaille and Andy Sutton were on the losing side for Zurich.
Czech Republic defender Pavel Kubina was fined a record 200,000 koruna (6,700 euros) and banned for 15 games Wednesday for criticizing a referee.
Kubina, a former NHL player who led Vitkovice to the Czech Elite league playoffs, said referee Petr Bolina was bribed and put on an unprofessional performance when officiating the seventh game of the semifinals between Vitkovice and Zlin.
Vitkovice, 3-1 up in the series before losing three straight games, conceded several goals during power plays in the decisive match. Kubina disputed the power plays and said Bolina was to blame for his team's elimination.
The fine was a record in Czech ice hockey, and the head of the Elite League said he would ask the Czech ice hockey association not to include Kubina in the team for the upcoming world championships in Austria.
"We have never dealt with a similar case," Stanislav Sulc was quoted by the CTK news agency as saying.
Kubina, who won the Stanley Cup with the Tampa Bay Lightning last year, has three days to appeal the decission.
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