Boston's Chris Kelly fired a slap shot past unproven Washington goaltender Braden Holtby for the game-winner just 1:18 into overtime to give the defending Stanley Cup champions a 1-0 series lead.
"I never thought it was going to be an easy game, I could tell that Holtby was on right away," Thomas said after finishing with 17 saves to Holtby's 29.
"I was just trying to make sure to stay on my toes, when you don't have that many shots it can be difficult."
Elsewhere, the San Jose Sharks picked the perfect night to figure out the St. Louis Blues with an upset 3-2 double-overtime victory.
Not to be outdone, The Phoenix Coyotes prevailed 3-2 in overtime against Chicago while the New York Rangers enjoyed the only convincing victory, 4-2 over Ottawa.
Despite losing all four meetings during the regular season, the seventh-seed Sharks fought their way into two overtimes before Martin Havlat put in the game-winner in the second session.
"It was a great pass from (Ryane) Clowe (in the slot), I just wanted to get it on the net and it went in," Havlat, who had two goals on the night, said.
"Everyone wants to have a chance to fight for the Cup, I'm just trying to enjoy it."
Patrik Berglund netted twice in the third to put the Blues up 2-1 but Andrew Desjardins tied it to force overtime.
Jaroslav Halak, who along with Brian Elliott forms the top goaltending unit in the NHL, made 31 saves but was upstaged by San Jose's Antti Niemi, who had 40 stops.
Quality goaltending was a common theme on Thursday as Mike Smith made 43 stops for Phoenix to top the 31 managed by Chicago's Corey Crawford. Martin Hanzal's winning tip-in sealed the game.
In New York, the Rangers played like the top seeds they are.
Having won the Eastern Conference for the first time since their 1994 Stanley Cup triumph, the Rangers looked ready to go all the way this year as they raced to a 4-0 lead and got 30 saves from Henrik Lundqvist.