Ali Price scored two tries and Wes Goosen one, giving the capital side a two-point lead at halftime, but Joe Edwards and Robin Taccola scored to keep it close.
Duhan van der Merwe scored after the break, and while Tani Vili crossed for Vannes, Ross Thomson’s penalty put the visitors two points ahead at the top of Pool 3. Sean Everitt’s team will face Georgian outfit Black Lion at home next Sunday.
Edinburgh started strongly and gained an early lead, pounding the Vannes line before Goosen scored on a superb long pass from Price. Darcy Graham broke through and dragged the cover defence before feeding the scrum-half to score, which benefited Price.
The French had to answer, and they did, with Edwards powering over from close range. Edinburgh made an outstanding effort to retake the initiative. James Lang created the break, exchanged passes with Graham, then found Price on a superb support line to complete a fantastic play.
The momentum shifted one way, then the other, and Vannes was quickly back in the game.
Salesi Rayasi burst through several weak tackles to get the hosts into the red zone, where the ball was recycled and passed wide to Taccola, who walked over untouched.
Jules Le Bail converted a penalty early in the second half to put Vannes in lead for the first time. However, substitute Thomson found Van der Merwe with a brilliant crossfield kick, and the massive winger over for the touchdown, regaining the lead for Edinburgh.
Their good work was undone by further slack defending, as replacement Vili stormed through multiple tackles to score and reduce Edinburgh’s advantage to one.
Thomson calmed the nerves with a penalty, and while Edinburgh had some tense moments in the final minutes, they held on for an important win.
Vannes: Duplenne, Rayasi, Taccola, Saili, Nakosi, Cotarmanac’h, Le Bail, Moukoro, Beziat, Tafili, Van der Merwe, Mezou, Boulier, Uhila, Edwards.
Replacements: Leafa, Djehi, Bourgeois, Desjeux, Parete, Varney, Vili, Ayarza.
Edinburgh: Goosen, Graham, Currie, Lang, Van Der Merwe, Healy, Price, Schoeman, Cherry, Sebastian, Sykes, Skinner, Ritchie, Crosbie, Muncaster.
Replacements: Harrison, Venter, Rae, Young, Watson, Shiel, Thompson, Tuipulotu.