Defenceman Petteri Nummelin scored two power-play goals in Finland's 6-2 win over Denmark, and forward Lee Stempniak scored twice to lift the United States over Belarus 5-1.
As in the 8-2 thumping of Belarus on Friday, the Czechs were ruthless on the power play, converting three times.
Peter Sykora and Hlinka scored in the first period, and the Austrians limited the Czechs to one goal in the second, by Petr Caslava on a power play at 10:02.
Raimund Divis ruined Roman Cechmanek's shutout bid when he wristed the puck from the sideboards over Cechmanek's left shoulder at 17:08.
In the final period, Czech goals went to Michal Barinka, Hlinka, and Jan Marek.
The Czechs, bidding for their third consecutive appearance in the final, will face their first real test of the tournament on Tuesday against the United States.
"That will be a tough game, our first real game here," Hlinka said.
The Americans were 3-0 up by the end of the first period against Belarus, with scores to defenceman Jack Johnson, and forwards Stempniak and Chad Larose.
Despite the second period becoming a little edgier as Belarus became more frustrated, Chris Clark added a fourth U.S. goal 4:30 in, and while enjoying a two-man advantage, Stempniak banged in the fifth - his third goal in two games.
Belarus captain Oleg Antonenko delighted the largely partisan crowd when he finally ruined John Grahame's shutout bid, deking defenceman Brian Pothier to flick the puck in over Grahame's shoulder.
Nummelin, playing in his 13th world championship for Finland, beat Denmark goaltender Michael Madsen with a wrist shot into the top corner on a 5-on-3 at 7:36 in the first period of the Group D game.
Denmark tied the score at 12:49 on another 5-on-3 power play through Jesper Damgaard's low shot from the point.
Mikko Koivu gave the 2006 Olympic silver medallists a 2-1 lead two minutes later with a short-handed goal, and Nummelin's second of the game on another 5-on-3 with just over a minute left in the opening period made it 3-1.
Peltonen scored Finland's third 5-on-3 goal at 3:28 of the second, defenceman Aki Berg scored early in the third, and Jukka Hentunen added another midway through the final period to complete the scoring for Finland.
Denmark got one back on a power play at 5-1 down when Christoffer Kjaergaard took a perfect centring pass from Frans Nielsen and redirected the puck.
Finland will face a much bigger challenge in its next game when it plays host Russia on Tuesday.
"Russia will be a big challenge for us," captain Ville Peltonen said. "They have a lot of skilled players and it will take a complete effort. The first two games have been pretty one-sided so this will be a good test for us."
Austria and Belarus will be in a must-win situation on Tuesday. The loser will play the relegation round in an attempt to avoid dropping out of world hockey's elite group. When the pair met at last year's worlds, Belarus won 5-0 and finished sixth.