Matthews, of the US, broke a scoreless tie in the third and Toronto goaltender Frederik Andersen made 28 saves including several late when Boston had pulled their goalie for an extra attacker.
David Krejci scored the Bruins' only goal with the Boston net empty but they couldn't get the second one to tie it up.
"This was his best 200 foot game of the playoffs," Leafs coach Mike Babcock said of Matthews. "He was outstanding."
The Maple Leafs can get their first NHL playoff series win in 15 years and advance to face the Columbus Blue Jackets in the next round with a victory in game six Sunday in Toronto.
With the teams locked in a scoreless tie after two periods, Toronto's Jake Muzzin found himself alone at the blue line, skated to the left circle and passed to Matthews, who notched his fourth goal of the playoffs from the right faceoff circle at 11:33.
The goal was upheld and Toronto's 1-0 lead stood.
"I had so many taken back that I got one back tonight," Matthews said of the fortunate call in the Leafs' favour.
Krejci scored at 19:16, but Toronto killed the final 44 seconds for the win.