The Czech side looked set for a dream start to life among Europe's elite when Marek Bakos, who scored six in qualifying, put them ahead on the stroke of half-time.
But Bressan latched on to Filipp Rudik's pass 21 minutes from the end to ensure the spoils were shared in Prague.
Plzen were immediately looking at home and they were first to threaten when David Bystron headed Pavel Horvath's corner over the bar.
At the other end, Dmitri Baga curled the ball towards goal but it was an easy save for Marek Cech.
BATE goalkeeper Aleksandr Gutor was given an uncomfortable moment when a free-kick from Bakos struck the wall and headed dangerously towards goal, but he recovered to save.
Gutor was in the action again after 15 minutes when he made a good low stop to deny Daniel Kolar.
Rudik's shot never looked like worrying Cech as it soared over the bar three minutes later, while Mateja Kezman fired wide from the left channel.
BATE were on top now and they rained shots in on goal, Edgar Olekhnovich shooting over and Kezman finding the side netting.
But Plzen, who were dominating the possession without turning it into chances, took the lead on the stroke of half-time when Petr Jiracek's low cross found Bakos, who tapped home.
Pavel Horvath tried his luck with an ambitious effort from distance two minutes after the restart and came close to doubling his side's lead as it went narrowly wide.
Kezman's snapshot two minutes before the hour mark went wide, and a minute later he had the ball in the back of the net but was denied by the offside flag.
Gutor kept his side in the game after 62 minutes when he saved from Vaclav Pilar.
And that proved crucial when Bressan ran on to Rudik's sliderule pass and slid the ball between the legs of Cech.
The game fizzled out somewhat thereafter as both sides seemed content to settle for a point.