The 30-year-old came out of a tangle along the right-wing boards for a short breakaway and roofed a shot past home goaltender Marc-Andre Fleury.
Ryan had 13 goals and a career-low 25 points in 62 games during the season but he now has five goals and 10 points in 13 playoff games.
“When you’re an offensive player and you’re doing all the right things but you’re not finishing and that’s what it was with Bobby...,” Senators coach Guy Boucher said.
“He just wasn’t finishing and it gets to be a mental block and you start thinking you’re not doing the right things. For me, now it’s the finishing part.”
Craig Anderson finished with 27 saves for the Senators, who improved to 6-1 in overtime in these playoffs while the Penguins dropped to 1-1.
Game Two is in Pittsburgh on Monday, when the defending Stanley Cup champions will be hoping to even up the best-of-seven series before it heads north of the border.
The Penguins started the series leading the playoffs with 41 goals, an average of 3.4 per game.
They also have the leading scorer in the playoffs in Evgeni Malkin, who began the night with 18 points, and four of the top seven.
But Ottawa stood firm through more than two periods and killed off all five Pittsburgh power-plays during regulation.
“We had an opportunity to grab a lead,” Penguins coach Mike Sullivan said. “We didn’t execute. The movement wasn’t there. The passes weren’t crisp.”
Pittsburgh had 5:50 of power-play time in the first period, including 45 seconds with a five-on-three advantage, yet it was Ottawa that scored first.
From behind the Penguins’ net after a turnover by Pittsburgh defenseman Brian Dumoulin, Ryan sent a soft backhander out front to Jean-Gabriel Pageau, who one-timed the puck past the glove of Fleury at 14:32.
It was Pageau’s eighth goal, one behind playoff leader Jake Guentzel of Pittsburgh.
Pageau had 12 goals during the regular season.
The assist was Ryan’s seventh of the playoffs after he had 12 during the regular season.
The Senators nearly claimed a 2-0 lead at 12:48 of the second on a wraparound play by Alexandre Burrows.
The puck found a way past Fleury at the right post but officials ruled the whistle blew first.
Malkin tied it 1-1 at 14:25 of the third on a tip-in from Chris Kunitz for his 19th point.
It also lifted Malkin past Jaromir Jagr into third place on Pittsburgh’s all-time playoff points list with 148.
That ruined Anderson’s shot at his second shutout of the playoffs.
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