Joakim Lindstrom was the defending champions’ hero as he scored the winner in a shootout, having also already scored and provided an assist.
The Czechs had led the Group A clash 3-1, but Sweden were not to be denied as Oscar Moller and Lindstrom scored power-play goals.
Tomas Hertl put the Czechs in front just eight seconds in before the Swedes found their feet and eventually prevailed courtesy by holding their nerve in the shootout.
Linus Klasen and Lindstrom kept their cool to score in the shootout and misses from Jan Kovar and Roman Cervenka proved to be costly for the Czechs as they suffered an agonising defeat.
Lindstrom said: “I think this is a huge win for us,
“They gave us every reason to give up and stop playing, but we never quit. We kept working and grinding and digging in there, and we found a way to win.”
Norway were also made to work for their second win of the tournament as they edged out Denmark 4-3 in Group A.
The Norwegians trailed 2-0 in the first period, but rallied to deny their fellow Scandinavian opponents.
Mats Trygg scored the winner as Norway enjoyed a numerical advantage during the third period to stun Denmark, who were made to pay for poor discipline as they surrendered four powerplays and a penalty shot in the third period.
Russia started their Group B campaign with a 4-2 win over Finland, while Belarus beat Kazakhstan 4-1 in the same group and Germany edged out Latvia 3-2.
Markus Gander’s late goal gave Italy a 2-1 victory over France in Group A.