Just like the other five games in the series, Toronto scored first and took a 1-0 lead in the first period when Ashton scored 8:45 into the game.
The goal came after Jamie Oleksiak turned the puck over behind Texas net. Cristopher Nilstorp made the initial save on T.J. Brennan, but Ashton was able to track down the rebound and scored his third goal of the post-season.
Toronto then doubled its lead after an ill-advised penalty against Nilstorp in the second period.
After a scramble by the Stars net, Nilstorp gave Toronto captain Trevor Smith a two-handed shove near the side of the net and was called for roughing. On the ensuing power play, Smith beat Nilstorp through the five-hole after taking a pass from Sam Carrick.
Ashton added his second goal with 54 seconds remaining in the period, tapping home a pass during a 2-on-1 with Spencer Abbott.
Toronto's success on the penalty kill helped extend the series. Texas had eight power plays, including a lengthy 5-on-3 in the second period, but didn't find the back of the net until Curtis McKenzie scored during a 6-on-4 advantage with 3:25 remaining in regulation.
Nilstorp stopped 24 of 25 shots in the loss.
Toronto has now scored first in all 13 of its playoff games.
Game 7 will be Tuesday night at the Cedar Park Center. The winner will play the St. John's IceCaps or Wilkes-Barre/Scranton Penguins in the Calder Cup Final.