Tour leader Chris Froome (Team Sky) was counting another successful day after finishing sixth and looked ahead to today’s finishing ascent to Mende.
Sagan (Tinkoff-Saxo), though, was fuming after Van Avermaet (BMC Racing) held him off to claim victory in stifling heat in south west France.
Slovakian Sagan, who has not won a stage for two years, said: “I’m not sad. I’m p****d. It’s different.
“It was my mistake. I was waiting too long to start my sprint.”
The 25-year-old is in the first year of three of his lucrative contract with Tinkoff-Saxo, but such is owner Oleg Tinkov’s frustration with results, it is rumoured he could be allowed to leave.
There is a suggestion Sagan could go to Etixx-QuickStep, with Mark Cavendish’s future at the Belgian squad uncertain.
It was the fourth time in the 2015 Tour that Sagan has finished second and 15th in four Tours. His most recent stage win came on the seventh stage in 2013, yet he has won the points classification in each of the last three races and is in possession of the green jersey again.
He added: “I’m happy to have some points, but for first place it would be more.
“If I continued pushing, maybe it could be different. That’s why I am a little bit angry now.”
The crash which saw Jean Christophe Peraud (Ag2r La Mondiale) tumble to the tarmac, tear his Lycra shorts and expose more than what was appropriate before the watershed was a reminder to Froome of the daily perils in the bunch.
And when the peloton was concerned it would not reel in the day’s breakaway, the day became “crazy”, Froome said.
The 2013 champion said: “What started off as quite an easy, steady stage turned crazy by the end.
“The heat was a big factor, with temperatures up close to 40C. We were just trying to send guys back every 10, 15 minutes to get bidons (water bottles). It still felt like it wasn’t enough.
“A 500m climb in the final like that, to see gaps opening up the way they did already gives us a little bit of an indication of what we can expect tomorrow at Mende.”
Today’s 14th stage, the 178.5km from Rodez to Mende, finishes with a three-kilometres climb before a run-in along an airstrip to the line and Froome expects further attacks on his advantage of two minutes 52 seconds from Tejay van Garderen (BMC Racing).
“It’s going to be a very selective day,” added Froome.
“That’s going to be really tough. More than 10 per cent for 3km.
“We can definitely expect it to be another GC (general classification) day again.”