Since gaining promotion to the Premiership at the end of last season, Welsh have struggled in their new surroundings, losing all six games in the top flight, conceding 272 points in the process.
That start was exacerbated last Saturday as they went down 28-18 at Lyon in their opening Pool Four encounter.
With Bordeaux, currently third in the Top 14 table, also losing their opening clash - 15-13 at home to Edinburgh - Burnell sees Thursday's meeting at the Kassam Stadium as a chance for Welsh to register their first win.
"We've got to get a win and that's what we'll be fighting for wholeheartedly to get - we have to get a win at home," he said.
"They're an excellent team and they've picked up some fantastic results so far this season, but we're back at the Kassam Stadium and we've got to get a win. We have to get a win for our loyal band of supporters.
"When people ask me how the mood in the camp is, I say 'it's good; they're up for the challenge and they're positive'. People must look at me a bit strange, but it's true.
"Of course we're disappointed, but it's a good place to be. We're giving it our best shot and these boys are dying for the cause."
Elsewhere in the pool, Edinburgh host Lyon, while in Pool Three Stade Francais - runners-up of the tournament in its previous guise in 2012-13 - travel to Bucuresti and Newport Gwent Dragons welcome Newcastle.
Exeter Chiefs have made a strong start in the Premiership, winning four of their first six outings, but they went down 30-24 at Bayonne on the Challenge Cup's opening weekend.
The Chiefs face Connacht - who themselves have made a promising start in the Pro12 - at Sandy Park while the other fixture in Pool Two sees La Rochelle meet Bayonne in an all-French affair.
Two sides who won their opening fixtures go head-to-head in Pool Five as Oyonnax welcome Gloucester.
Oyonnax are rock bottom of the Top 14, while Gloucester have been inconsistent in the Premiership thus far, winning three and losing three of their six matches. Brive and Zebre meet in the pool's other game.
Elsewhere, in Pool One, London Irish - fresh off the back of their 70-14 hammering of Rovigo - travel to Grenoble, while Cardiff Blues visit the Italian outfit looking for another five-point haul.