Pastrnak raced up the right wing and scored off the carom of defenseman Hampus Lindholm's lob off the corner boards, tucking a backhander between the left post and the skate of Toronto goaltender Ilya Samsonov (29 saves).
Lindholm finished with a goal and an assist, tallying Boston's only marker in regulation just 1:21 after Toronto's William Nylander opened the scoring at 9:01 of the third period.
Jeremy Swayman made 30 saves for Boston, who have now won seven consecutive playoff series against Toronto dating back to 1969 and will meet the Florida Panthers in the second round.
Game 1 of the Bruins' next series is set for Monday night in Sunrise.
Samsonov got the start since Joseph Woll did not dress due to an undisclosed injury sustained in Game 6.
Nearly midway through the third, Toronto scored the go-ahead goal off an in-zone sequence. After Boston defenseman Brandon Carlo slipped and lost the puck in the corner, Auston Matthews - who returned from a two-game absence (illness) - picked it up and dished it to an open Nylander for him to deposit into the back of the net.
The hosts wasted little time responding. Justin Brazeau's backhand from the bottom of the right circle deflected over the net, but Lindholm got the puck on the opposite wing and flicked it in off the right post.
Boston combined for just three first-period shots in Games 5 and 6, but it got off to a much better start on Saturday, outshooting the Maple Leafs 11-9 through the first 20 minutes of action.
However, Toronto picked up the pace in the second, recording 12 shots to the Bruins' seven.
Following a scoreless Boston power play, Toronto's first and only man advantage - stemming from a Charlie Coyle cross-check at 4:36 of the second - resulted in strong pressure. The Bruins completed the kill, with Tyler Bertuzzi putting a shot on goal just as the power play was coming to an end.
Another key Swayman save came on Connor Dewar's short-handed breakaway with 8:35 left in the period.
