The busy Giroux added three assists to his tally with rookie Couturier also grabbing one for the Flyers, who came back from a 3-1 first period deficit to win 8-5 and take a 2-0 lead in their best-of-seven Eastern Conference series.
Elsewhere, New Jersey spoiled Florida's return to the postseason and Detroit evened their series with Nashville.
The visiting Flyers trailed 3-0 in Game One before winning 4-3 in overtime and this time, Philadelphia scored the game's final four goals, two of them by Couturier, to overturn Pittsburgh's 5-4 lead at the start of the third period.
Couturier, 19, became the first teen-ager to score a hat trick in a play-off game since Ted Kennedy in 1945.
"If he didn't score any goals, I would say he had a great game," Philadelphia's Jaromir Jagr said of team-mate Couturier.
"But he played great defense and also scored three goals. I don't know if I know any words to describe his game today. 'Awesome!' Maybe something better than that. He was unbelievable."
Giroux's empty netter closed out the scoring with seven seconds remaining as he scored a franchise record six points, leading the Flyers to a pair of road wins before the series shifts to Philadelphia for Game Three on Sunday.
Sidney Crosby's goal 15 seconds into the contest helped spark a then-dominant Pittsburgh to a 3-1 lead, but the Flyers stood firm, then increased their intensity and were rewarded with three second-period goals to tie the game at 4-4.
Pittsburgh did not score after Tyler Kennedy's backhand 1:04 into the third had put them back ahead.
"We just need to find a way to win, close out games, and play a full 60 minutes," said Pittsburgh James Neal. "It's frustrating when we have such a good first period and we let it slip away."
In the West, Jonathan Quick made 46 saves and Dustin Brown scored two short-handed goals to give the surprising Kings their second consecutive road win over the league's top regular season team.
Third period scores by Jarrett Stoll and Trevor Lewis made it 4-1.
The best-of-seven series moves to Los Angeles for Game Three on Sunday.
Meanwhile, Ian White, Cory Emmerton and Johan Franzen scored for Detroit in a 3-2 victory as the Red Wings evened up their Western Conference series against the Nashville Predators at 1-1.
Game Three is in Detroit on Sunday.
In the other Eastern Conference game, Dainius Zubrus and Ryan Carter scored 45 seconds apart in a three-goal first period to lead the New Jersey Devils a 3-2 win over a Florida Panthers team making their first play-off appearance in 12 years.
Martin Brodeur made 24 saves for his 100th play-off victory.
Game Two is also in Sunrise, Florida on Sunday.