The 25-year-old Matthews, with three Vuelta and two Giro d’Italia stage wins to his name, outsprinted his breakaway companions after 197km from Escaldes-Engordany, Andorra, benefiting from the support of two of his Orica-Bike Exchange teammates on Tuesday.
World champion Peter Sagan (Tinkoff), who reclaimed the green jersey for the points classification from Briton Mark Cavendish, took second place and Norway’s Edvald Boasson Hagen (Dimension Data) finished third.
Matthews missed the 2014 Tour after sustaining injuries in training and, last year, he abandoned after breaking four ribs.
“And this year already I had crashed twice. I was almost giving up on this race, thinking maybe the Tour de France is not for me,” he said.
“But today my dream has come true.”
Briton Chris Froome (Team Sky) retained the overall leader’s yellow jersey after finishing safe in the main peloton 9:39 off the pace.
“The start was pretty full on, there was a lot of work for my teammates to control the race,” said Froome.
“But it was pretty straightforward from there. However, it’s a good day to tick off.”
Australia’s Richie Porte remains two minutes and 10 seconds behind Froome.
“To be honest, I’m just happy to get back into the swing of things and be racing again after the rest day,” Porte said.
“There’s a flat stage tomorrow and then well be back on the climbs.”
The stage got off to a nervous start on the category-one climb to the Port d’Envalira, as riders were keen to win places in the day’s breakaway which took its final shape after about 60 kilometres.
It was a royal break as six of the 15 fugitives had already worn the yellow jersey and nine of them were Tour stage winners.
With 24km left, Sagan’s brutal acceleration on a small rise split the group and only seven, including two of Matthews’s teammates, remained in front.
Matthews benefited from compatriot and teammate Luke Durbridge’s work to keep the chasers at bay and then stayed on Sagan’s wheel as his other teammate, South African Daryl Impey, led the group in the lead-up to the final sprint.
Impey also attacked on the short climb up the Cote de Saint Ferreol close to the finish, forcing Sagan to make efforts that probably left the Slovak short of juice in the finale.
Matthews then delivered with an impressive burst of speed over the final 200 metres.
“I would not have won without Luke Durbridge, and Daryl Impey attacking in the climb and leading out the sprint,” said Matthews.
Wednesday’s 11th stage is a 162.5-km ride between Carcassonne and Montpellier that should favour the sprinters before Thursday’s trek up to the intimidating Mont Ventoux.
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