The goal marked Ryan's fifth of the playoffs and rewarded a strong defensive performance by the Sens, highlighted by goaltender Craig Anderson stopping 27 of the 28 shots he faced.
Sportsnet provided a look at Ryan's game-winner on a partial breakaway:
Evgeni Malkin scored with 5:35 left in the third to force overtime, but Ottawa's tight-checking style was successful, as it held Penguins captain Sidney Crosby to just two shots on goal.
Saturday's victory marked the Sens' sixth overtime win of the 2017 playoffs.
Pittsburgh largely dominated the first period in terms of puck possession and quality chances, as the Senators were called for four minor penalties.
Although Pens winger Patric Hornqvist rung a shot off the post on one of Pittsburgh's power plays, the Ottawa penalty kill was flawless.
The Sens gained momentum by killing off a five-on-three power play and went on to open the scoring 14:32 into the game, when forward Jean-Gabriel Pageau beat Penguins goalie Marc-Andre Fleury after a picture-perfect, no-look pass by Ryan, as seen in this video courtesy of the NHL's official Twitter account:
That goal marked Pageau's eighth of the postseason, and it continued his career-long trend of playoff excellence, according to TSN's Brent Wallace:
Ottawa once again found itself on the penalty kill after an interference call on Mark Stone 7:33 into the second period.
The Penguins continued to press for the tying goal, but Anderson was on top of his game, making saves such as this remarkable glove stop on defenseman Justin Schultz's tipped point shot:
Anderson preserved the 1-0 lead, and Ottawa nearly added to it with seven minutes remaining in the middle frame.
Forward Alexandre Burrows put the puck past Fleury on a wraparound, but it was ruled no goal because of the official having intent to blow the whistle after losing sight of it.
Kurt Ludlow of WSYX in Columbus, Ohio, was among those who disagreed with the decision:
Despite the controversy, Ottawa held the advantage entering the third period, as it largely stymied Pittsburgh during five-on-five play.
As pointed out by Murray Pam of the Hockey News, the Penguins struggled mightily to set up shop in the offensive zone:
The Penguins possessed the puck for much of the first 14 minutes of the third period, with Ottawa seemingly content to stay on defense.
That strategy appeared to be working until Pittsburgh finally broke through when Chris Kunitz blindly threw a shot toward the net and Malkin deflected it past Anderson to tie the game at 1-1:
Malkin made the play happen with remarkable hand-eye coordination, prompting Dimitri Filipovic of Sportsnet to sing the Russian superstar's praises:
With that marker, Malkin further solidified his place in Penguins playoff history, according to Josh Getzoff of PensTV:
Phil Kessel hit the crossbar shortly after Malkin tied it up, but Ottawa was able to escape the period, and the teams headed to overtime.
Ottawa didn't allow Pittsburgh's late goal to swing the momentum too greatly, as Ryan continued his impressive playoff performance with his third game-winning goal of the postseason.
The Senators won each of their first two series in six games, and they were a slightly fresher team Saturday after the Washington Capitals took Pittsburgh to seven games last round.
Ottawa has made a habit of shutting down the opposition's top scorers while finding timely scoring of its own, and it will need to continue that trend for the rest of the series to upset the high-powered Pens.