Van Avermaet sat at just 16 seconds down on race leader Peter Sagan at the start of the stage, with the world champion's team only able to come home with a deficit of 50 seconds to BMC and Sagan himself dropped in the latter part of the course. Van Avermaet finished ahead of his team-mate Tejay van Garderen to move into yellow with the pair on the same time at the start.
The American team clocked a fastest time of 38-46 over the 35.5km course around Cholet, four seconds faster than Team Sky who had sat in the lead briefly before being beaten.
The only team to look like threatening BMC in the hot seat were Quick-Step Floors, who went through the second intermediate split at just six seconds down, but were unable to make up the ground and eventually finished third at seven seconds.
Richie Porte and Sky's Chris Froome and Geraint Thomas will all finish the day satisfied with their gains in GC. Thomas is the best placed rider sitting in third place at three seconds behind Van Avermaet and van Garderen respectively. Porte sits at 51 seconds, while Froome is back at 55 seconds overall.
Mitchelton-Scott, who had gone off first on the stage, will be satisfied with keeping their leader Adam Yates in touching distance of the likes of Porte, Froome and Thomas, having finished fourth at just nine seconds on the winners and five on Sky.
Former world champions at the discipline Team Sunweb also managed to keep their leader Tom Dumoulin in contention, finishing in fourth place at 10 seconds and leaving the Dutchman 11 seconds behind Van Avermaet on GC.
Movistar may end the day thankful that Alejandro Valverde and Mikel Landa took early gains on some of their rivals in the first two stages, with the Spanish team finishing 53 seconds back and dropping to 53 seconds behind on GC as well. They remain amongst a clustered group of GC contenders however, just ahead of Froome.
Nairo Quintana has a huge amount of ground to make up now though, sitting 2-08 down on the overall lead and 2-05 down the best placed of the GC hopefuls in Welshman Thomas.
Likewise Ag2r La Mondiale's Romain Bardet will be thankful he made the front group on stage one, as his time buffer on some of the favourites was obliterated after the French team came across the line at 1-15 down on BMC.
Dan Martin expectedly finished the day with a big chunk of time to make up after his UAE Team Emirates squad came in 1-38 down. With a time trial coming at the end of the race, the Irishman will need to make significant gains in the mountains if he stands any chance of taking victory.
Of the other contenders, Vincenzo Nibali and Bahrain-Merida finished at 1-06 down after looking like they would mitigate any losses in the early stages, while EF-Education put in a fine ride for Rigoberto Uran at 35 seconds, with the Colombian now in the top-10 overall.
The Tour de France continues on Tuesday with stage four, a flat 195km route from La Baule to Sarzeau.
Tour de France 2018, stage three: Cholet - Cholet (35.5km, TTT)
1 BMC Racing Team, at 38-46
2 Team Sky, at 4s
3 Quick-Step Floors, at 7s
4 Mitchelton-Scott, at 9s
5 Team Sunweb, at 11s
6 EF Education First-Drapac, at 35s
7 Bora-Hansgrohe, at 50s
8 Astana Pro Team, at at 51s
9 Katusha-Alpecin, at 52s
10 Movistar Team, at 53s
11 Bahrain-Merida, at 1-06
12 AG2R La Mondiale, 1-15
13 Team LottoNl-Jumbo, at same time
14 Trek-Segafredo, at 1-16
15 UAE Team Emirates, at 1-38
16 Groupama-FDJ, at 1-42
17 Fortuneo-Samsic, at 1-46
18 Direct Energie, at 1-51
19 Lotto Soudal, at same time
20 Dimension Data, at 1-52
21 Wanty-Groupe Gobert, at 2-24
22 Cofidis, Solutions Credits, at 3-23
General classification after stage three
1 Greg Van Avermaet (Bel) BMC Racing Team, in 9-08-55
2 Tejay van Garderen (USA) BMC Racing Team, at same time
3 Geraint Thomas (GBr) Team Sky, at 3s
4 Philippe Gilbert (Bel) Quick-Step Floors, at 5s
5 Bob Jungels (Lux) Quick-Step Floors, at 7s
6 Julian Alaphilippe (Fra) Quick-Step Floors, st
7 Tom Dumoulin (Ned) Team Sunweb, at 11s
8 Søren Kragh Andersen (Den) Team Sunweb
9 Michael Matthews (Aus) Team Sunweb, all same time
10 Rigoberto Uran (Col) EF Education-First Drapac, at 34s
13 Jakob Fuglsang (Den) Astana Pro Team, at 51s
14 Richie Porte (Aus) BMC Racing, at same time
15 Ilnur Zakarin (Rus) Katusha-Alpecin, at 52s
16 Alejandro Valverde (Esp) Movistar, at 53s
17 Mikel Landa (Esp) Movistar, st
18 Chris Froome (GBr) Team Sky, at 55s
20 Adam Yates (Gbr) Mitchelton-Scott, at 1-00
22 Vincenzo Nibali (Ita) Bahrain-Merida, at 1-06
25 Romain Bardet (Fra) Ag2r La Mondiale, at 1-15
39 Daniel Martin (Irl) UAE Team Emirates, at 1-38
59 Nairo Quintana (Col) Movistar, at 2-08
Thank you for reading 5 articles this month* Join now for unlimited access
Enjoy your first month for just £1 / $1 / €1
*Read 5 free articles per month without a subscription
Join now for unlimited access
Try first month for just £1 / $1 / €1