Szturc drove down the right wing and cut in around defender Denis Bakala before lifting a high shot over the glove of Simon Latkoczy. It came after Slovakia had scored twice on the power play to rally from 4-2 down late in the third.
The only reason the score was close was Latkoczy, who faced 52 shots and was the star of the game.
Jan Mysak and Michal Gut each had a goal and assist for the winners while Matej Kaslik had two goals for Slovakia.
"We're happy we have a win," Mysak said. "That's all we want. That's all we're focused on. And we have it, so it's a perfect start. The key thing for us is just to skate. When we have good movement, we have opportunities, right? So we just try to skate and shoot every single puck and have guys around the net to score goals."
Both teams now have a day off before playing again on Thursday. Czechia will play the early game against Finland followed by Slovakia taking on hosts Canada.
"I think we were good from the start," said Slovakian forward Servac Petrovsky. "We scored two goals. We played really well in the first period, but the second period was very bad for us."
The teams last played almost seven years ago and are in the same group this summer thanks to the suspension from participation of Russia, which forced a re-seeding of the groups after the cancelled Christmas edition of the tournament.
Two and a half minutes later, the Slovaks doubled their lead off the rush. Kaslik took a pass form Oleksiy Myklukha as they crossed the Czechia blue line, and Kaslik lifted a weak shot over Bednar’s glove that the goalie should have stopped.
It was a different Czechia team that came out for the middle 20 minutes. They got back into the game at 2:22 off a nice play from the side of the goal. Matous Mensik fed Martin Rysavy with a quick pass in front under the stick of goalie Simon Latkoczy to cut the lead to 2-1, and that started a period of near total domination. It was one of 24 shots Latkoczy faced, and he was the pnly difference between a close game and blowout.
Czechia tied the game midway through the period after captain Jan Mysak was stopped off the rush by Latkoczy. In the ensuing scramble, Michal Gut managed to jam the puck in to tie the game at 2. Four minutes later, Czechia took its first lead. Latkoczy took a high shot off the shoulder from Tomas Urban, and an unchecked Matyas Sapovaliv was right there to knock in the loose puck. Sapovaliv is the brother of Adela Sapovalivova, who was a star for Czechia at the Women's U18 this past June.
Mysak doubled the Czechia lead at 7:56 of the third period. He coralled a loose puck at the Slovakia blue line and went in alone, firing a shot past Latkoczy's glove for what looked to be the insurance marker.
Nope. Six minutes later, Slovakia earned a five-on-three for 68 seconds and wasted no time in capitalizing. Captain Rayen Petrovicky wired a hard shot over Bednar's glove from the top of the circle, cutting the lead to one. To make matters worse, David Spacek cleared the puck over the glass, earning a minor penalty and giving the Slovaks another two-player advantage for 18 seconds.
Sure enough, the Slovaks tied the score late in the second penalty. Tomas Hamara made a terrible clearing in front of his goal, and in the ensuing scramble Kaslik pushed the puck over the goal line, setting the stage for Szturc's late-game heroics.
"Our team spirit showed up," Dvorsky said of his team's comeback in the third. "We didn't give up. The atmosphere was really good on the bench. We were pushing forward and we scored two goals, so that's really good."