Both teams will have to head back to Salamanca again next week for the deciding game and that will suit Bourges just fine after this performance and win that would have been more comfortable but for a big Anke De Mondt triple with three-seconds remaining.
That brought Halcon Avenida to within three points but Anael Lardy sunk the winning free throws and got the home fans on their feet.
Bourges controlled much of the game and they always looked the more likely winners after opening up with a 20-15 advantage after the first quarter and looking relatively comfortable at 38-29 heading into the second half.
However Halcon Avenida were always likely to make a comeback and that came instantly after both teams emerged from the locker rooms.
The Spanish side started with an 8-0 run to get within a point but it was to be as close as they would get as the home team regained their composure to chalk up a vital win.
While ending up on the losing team, the outstanding Sancho Lyttle produced an MVP performance as she mustered a wonderful 20 points and 16 rebounds.
Greek star Styliani Kaltsidou top scored for Bourges with 16 points and Emmeline Ndongue weighed in with 13 points and 7 rebounds.
Pierre Vincent, Bourges Basket coach:
"In the first game Salamanca played excellent offensively. The Salamanca team are strong but they are also only humans so we can tire them if we play hard enough. We played a good game but we still have to be better in our head. Had we been stronger tonight we could have won by 10-15 points. Now we have nothing to lose and everything to play for."