Van Avermaet now holds a two minute, 22 second lead over British rider Geraint Thomas.
It was the first stage of this year's Tour to hit the alps, and Greg Van Avermaet extended his lead in the yellow jersey, as Julian Alaphilippe took out the stage. "When they go, it is never nice to see them all riding away but we had confidence in each other and we rode really well".
Sky teammate Geraint Thomas is second over all, 43 seconds behind van Avermaet, meaning the team has two viable options to claim the yellow jersey over the second week of the three-week Tour.
But Team Sky kept their powder dry until Thomas launched his move.
The 109-mile stage features three beyond-category climbs - Col de la Madeleine, Croix-de-Fer and the iconic Alpe d'Huez finish after 21 switchbacks to close out the Tour's three days in the Alps.
Froome himself would subsequently follow a move from UAE Team Emirates rider Dan Martin and finished third, crossing the line just behind Dumoulin, both riders 20 seconds behind Thomas. I knew it was a good chance but I didn't know what everyone else would ride like. "Froomey's won six Grand Tours".
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With van Avermaet not a specialist climber, Thomas could potentially take yellow today when the Tour resumes with a huge 158.5km mountain stage from Annecy to Le Grand Bornand.
The top five in the overall classification is rounded out by 2014 champion Vincenzo Nibali in fourth, 2:14 behind Thomas, and Primoz Roglic in fifth, 2:23 back.
Alaphilippe now leads the King of the Mountain standings and will wear the polka dot jersey during Stage 11. We were expecting attacks.
Thomas won the Critérium du Dauphiné heading into the Tour while Froome recovered from winning the Giro d'Italia.
Twice podium finisher Romain Bardet, hoping to become the first Frenchman to win the Tour since Bernard Hinault in 1985, has suffered from bad luck with four ill-timed punctures, including three in Sunday's stage to Roubaix.
Dumoulin then attacked on the descent of the penultimate climb, the Cormet de Roseland, linking up with Valverde before the road rose up again towards La Rosiere.