With the Maple Leafs trailing 2-1 in the third period, Armstrong beat Philadelphia goalie Sergei Bobrovsky to send the game to overtime.
Kris Versteeg assisted on the tying goal. The Leafs acquired Armstrong and Versteeg in the offseason to shake up an offense that had trouble scoring last season.
Greg McKegg also scored for the Leafs. Marc-Andre Bourdon and Bill Guerin scored for the Flyers.
The Leafs improved to 2-1-0 in the preseason while the Flyers fell to 1-0-1.
An enthusiastic crowd at London's John Labatt Centre was on hand to see local favorite Nazem Kadri, who is expected to crack the Maple Leafs' lineup this season.
Kadri, the Leafs' top draft choice a year ago, played on a line with Armstrong and Versteeg. He didn't get much room during the game from the tight-checking Flyers, but he did score the Leafs' first goal in the shootout.
Luca Caputi and Christian Hanson had Toronto's other shootout goals while Jeff Carter, Guerin and Joonas Lehtivouri scored for Philadelphia.
After a defensive first period in which Bourdon scored the only goal, the game opened up a little in the second period with both teams getting decent chances.
Versteeg made a nice play breaking in from the right wing when killing a penalty and forcing Flyers goalie Brian Boucher to make a good save.
The Leafs pressed hard during a power play and were rewarded when McKegg scored shortly after the Flyers returned to full strength. But the cheering hadn't stopped when Guerin put the Flyers ahead again with an unassisted goal.
Boucher was strong in the two periods he played, before giving way to Bobrovsky at the start of the third period.