Washington opened the playoffs by losing twice to the New York Rangers and had to rally from a 3-1 deficit to advance. The Capitals had no desire to have to do the same thing against Pittsburgh.
"This team is better," Capitals coach Bruce Boudreau said of the Penguins. "No disrespect to New York, but we'd have no chance being down 2-0 to be able to duplicate it."
Game 2 is Monday night in Washington.
Mark Eaton also scored for the defending Eastern Conference champion Penguins, who went 0-for-5 on the power play after going 1-for-18 in the final four games of their first-round series against Philadelphia.
"We need to do a better job on the power play," Crosby said. "We'll go over it and make sure we fix those things."
The Penguins and Capitals have met seven times previously in the playoffs; Pittsburgh has won six of those series. Seventeen of the 43 playoff games between the teams have been decided by one goal, including this thriller.
"Our shot total was 36. That's something you like to see at the end of the game," Pittsburgh coach Dan Bylsma said. "I think we could have added a few more. We had opportunities. We had chances to get some more goals. We didn't get enough to get the goals we needed to win, so we're going to try to get some more from the 36 shots."
With the score tied at 2 late in the second period, Varlamov twisted his body and stretched his stick across the goal line to deny Crosby, who had taken a shot from the left of the crease after receiving a cross-ice pass from Chris Kunitz. Crosby lifted his arms momentarily to celebrate before realizing the goaltender managed to keep the puck out of the net.
"I was out of position, there was really nothing I could do, so I lunged with the stick, and it just so happened the puck hit it right on the line," Varlamov said.
"It was kind of a desperation save," Crosby said. "You don't want to waste those opportunities."
Fleischmann got the only goal of the third period. Semin brought the puck into the zone and sent a diagonal pass to Nicklas Backstrom, who passed the puck across the ice to Fleischmann. That created an opening, and Fleischmann easily beat Marc-Andre Fleury for his second goal of the playoffs.
"I just went to the net, stopped and chipped it over," said Fleischmann, a second-line winger. "When your top guys have a lot of pressure on them, you have to take it away with secondary guys, like we are, and try to put it in."
Crosby put the Penguins up 1-0 with a wrist shot from between the circles at 4:09 of the first period. Pittsburgh dominated the first 10 minutes and built an 11-2 advantage in shots on goal before Steckel slammed in the rebound of a shot by Matt Bradley at 13:50.
It was the Capitals' first goal in May since Joe Juneau scored against Buffalo in the 1998 Eastern Conference finals.
At 17:03, Ovechkin scored from the left side after taking a pass from Semin with Washington working a 5-on-3.
Pittsburgh tied it at 12:54 of the second period when Eaton took a slap shot from just inside the blue line that slithered under Varlamov's glove and into the net. Varlamov redeemed himself later in the period with his outstanding stick-save against Crosby.
"He owed us one," Boudreau said.
Notes: Varlamov has played in more playoff games (seven) than regular-season games (six). ... Pittsburgh scratches included Miroslav Satan, who played in two games during the opening round. Washington scratches included suspended winger Donald Brashear.