In Moscow, Timo Parssinen scored two power-play goals in the second period as Finland opened with a 5-0 win over Ukraine.
"I thought it was critical that we get off to a good start," U.S. coach Mike Sullivan said. "Certainly, if you can win the first game it takes a little bit of pressure off as far as your ability to advance in this tournament.
"When you leave it up to chance or one game, it get difficult because you can run into a hot goaltender or something. I definitely reiterated to our players that this was a very, very important game for us."
On Sunday, the U.S. team faces Belarus and Austria plays the 2006 silver-medallist Czech Republic.
The United States is seeking its first medal since winning bronze at Prague in 2004. At the last Moscow worlds in 1986, the U.S. team lost in the preliminary round robin but avoided relegation.
Boston Bruins forward Phil Kessel, who helped the United States win world junior gold in 2005, scored first less than two minutes into the contest, coming out from behind the net to fire the puck over Reinhard Divis' glove.
The Oilers' Tobias Petersen doubled the score at 10:22 when the puck bounced off his skate and through Divis' five-hole. Raimund Divis finally put Austria on the board at 14:51, but Stempniak converted a two-man power play less than three minutes later to restore the two-goal lead.
Chris Clark of the Washington Capitals scored 24 seconds into the following period to make it 4-1. But Austria's Oliver Setzinger closed out the second-period scoring at 10:51 to narrow the gap.
Erik Cole and Brandon Bochenski put the game out of reach, each scoring in the final session.
"We needed to get one under our belt and we did," Nashville Predators defenceman Ryan Suter said. "We just want to win, and winning by the score we did was a bonus. We had it on them in the first period and then they turned it on a bit at the end of the second period. Hopefully, we can get another one on Sunday."
"It's pretty hard to play catch-up hockey with Team USA. They scored when they had to, early in every period," Austria coach Jim Boni said. "But we have to stay positive. We played a team full of skilled NHL players, a team that can contend for a medal. We're 16th in the world.
"Our goal is not to go down. It's going to be even tougher against the Czechs."
In Moscow, Parssinen deflected from the point a shot by Petteri Nummelin at 7:18 of the second period to give Finland a 3-0 lead.
He scored again on a 5-on-3 six minutes later when he picked up a rebound and tucked it behind Ukraine goaltender Igor Karpenko on a backhand.
Sean Bergenheim also scored for Finland in the second and Petri Kontiola completed the scoring with a power-play goal midway through the third.
Finland took a 1-0 lead at 12:07 of the first period.
Ville Peltonen of the Florida Panthers fed Pekka Saravo, who beat Karpenko with a slap shot from the high slot.
Fredrik Norrena, playing in place of the injured Antero Niittymaki, faced little pressure but made several key saves in the third period to preserve the shutout.