It was a day that many predicted would be too hard for van Avermaet, but the strong Belgian showed plenty of fight to not just keep, but extend his lead in the yellow jersey.
Starting the day with a 43 second advantage, van Avermaet extended his lead to two minutes and 22 seconds when the group containing the heads of state including Chris Froome (Sky) rolled across the finish three minutes and 23 seconds after the stage winner.
Geraint Thomas (Sky) remains in second place while Alejandro Valverde (Movistar) jumped a couple of places to move to third overall.
"It was a nice day I think," said van Avermaet. "I tried to do a thing I did a few years ago, going in the break with the Yellow Jersey and see how Sky and the other teams react. It seems they let me go and they gave me seven minutes so I was able to keep it. It was a super nice day, an extra day in yellow, in the mountains, so it was something special."
"One day, tomorrow I think I'm going to lose it," he joked. "I like it, I enjoyed what I did in Roubaix. Attacking with this Yellow Jersey is something cool and I don't want to just give it away.
Stage 10 at the Tour de France 2018, Yellow remains at BMC and what a race at 'La Course"
"Sometimes you have to race with the weapons you have, and that was what I did today. It's pretty cool to extend for another day and go for it."
Chris Froome is still the man to beat in the race for yellow and was diplomatic when asked if Team Sky could have pushed to take van Avermaet's Yellow Jersey from him on Stage 10.
"I don't want to take anything away from Greg Van Avermaet," said Froome. "He did a fantastic ride today. Getting himself into that break on such a mountainous day wasn't easy so he did a great job to hang on to it and I'm sure he'll be fighting for it again tomorrow."
That said, Froome went on to extoll the reasons it might have been an easier day for the Belgian star of the classics to keep a hold of the lead.
"No one really showed their cards today. Everyone played it a little bit conservatively. Maybe they were thinking about the next two days, which are going to be pretty hard as well. It wasn't a summit finish. I don't think our big rivals necessarily showed their hand today."
Stage 11 runs 108.5 kilometres from Albertville to La Rosiere, with coverage starting from 9.30 pm AEST on SBS and via streaming on Cycling Central and the SKODA Tour Tracker.