Matthews found himself among an elite 13-man breakaway shortly after the category one climb of Port d'Envalira, with Mikel Landa (Sky), Gorka Izagirre (Movistar), Vincenzo Nibali (Astana), Peter Sagan (Tinkoff), Samuel Dumoulin (AG2R-La Mondiale), Damiano Caruso (BMC), Edvald Boasson Hagen and Steve Cummings (Dimension Data), Tony Gallopin (Lotto-Soudal), Rui Costa (Lampre-Merida), and Orica teammates Luke Durbridge and Daryl Impey all there.
The opening of the stage was full of attacks as riders chased the KOM points on the summit of the climb and tried to get in the breakaway, with peloton looking happy to allow the break to go when the pace settled after the descent.
That allowed Sagan to take the intermediate sprint points ahead of Matthews in the stage, meaning the Slovakian will take over the lead of the points classification from Mark Cavendish (Dimension Data).
The break eventually built up an advantage of well over six minutes, and though that dropped to around just over four minutes, world champion Sagan's attack with 24km to go looked to ensure that the break would stay away.
That attack caused a split in the 13-man group, with only Matthews and his two Orica teammates, Boasson Hagen, Van Avermaet and Dumoulin able to follow.
Watch: Tour de France 2016 stage 10 highlights
The seven other riders quickly slipped to around a minute behind as the front group approached the final category three climb of the stage with around 9km to the finish, with seven minutes back to the peloton.
And that's where Orica attempted to make their numbers count. Despite the climb being only 1.8km in length, Impey's consistent attacks on the climb forced Sagan to work hard to chase as the others followed his wheel.
The front group eventually came together into the final straight on the 197km stage from Andorra, and it was Van Avermaet who tried to jump first. Matthews then launched himself on the right hand side of the road, with Sagan slightly crowded out as he tried to reply.
But Matthews was a bike length clear by the time he hit the line, and Sagan was able to hold on to second with Boasson Hagen in third.
There was no change to the overall standings after stage 10, with race leader Chris Froome kept safe by his Team Sky squad.
Wednesday sees a chance for the sprinters on a flat stage from Carcassone to Montpellier, with Mark Cavendish aiming for his fourth stage win of the race.
Tour de France 2016 stage 10, Andorra - Revel (197km)
1. Michael Matthews (Aus) Orica-BikeExchange in 4-22-38
2. Peter Sagan (Svk) Tinkoff
3. Edvald Boasson Hagen (Nor) Dimension Data
4. Greg Van Avermaet (Bel) BMC Racing
5. Samuel Dumoulin (Fra) Ag2r at same time
6. Daryl Impey (RSA) Orica-BikeExchange at 2 secs
7. Luke Durbridge (Aus) Orica-BikeExchange at 1-10
8. Damiano Caruso (Ita) BMC Racing at 3-01
9. Gorka Izaguirre (Esp) Movistar at 3-10
10. Tony Gallopin (Fra) Lotto-Soudal at same time
General classification after stage 10
1. Chris Froome (GBr) Team Sky in 49-08–20
2. Adam Yates (GBr) Orica-BikeExchange at 16 secs
3. Daniel Martin (Irl) Etixx-QuickStep at 19 secs
4. Nairo Quintana (Col) Movistar at 23 secs
5. Joaquim Rodriguez (Esp) Katusha at 37 secs
6. Bauke Mollema (Ned) Trek-Segafredo at 44 secs
7. Romain Bardet (Fra) Ag2r at 44 secs
8. Sergio Henao (Col) Team Sky at 44 secs
9. Louis Meintjes (RSA) Lampre-Merida at 55 secs
10. Alejandro Valverde (Esp) Movistar at 1-01
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