Forward Marek Bakos put the Czech champions, who won six out of six in the three qualifying rounds they played, ahead a few seconds before half-time from close-range after a perfect cross-pass from the left side by Petr Jiracek.
Plzen exerted early pressure but Borisov's defence rarely let them near a sight of goal while the Belarussian side wasted opportunities of their own.
"I am disppointed now right after the match but it probably was a deserved draw," said Plzen skipper and playmaker Pavel Horvath. "[But] I have said I would be happy with every point from the Champions league.
"We had some shooting positions that we should have resolved better, be more productive... maybe we made a few errors in passing that we normally don't do."
BATE striker Mateja Kezman, the most experienced man on the pitch who has played for a string of European teams including Chelsea, shot into the side netting on the half hour and had a goal ruled out for offside 15 minutes into the second half.
Plzen went close to increasing their lead in the 63rd minute but Vaclav Pilar, who impressed throughout, failed to capitalise on a pass into the box, firing a weak shot that was easily smothered by keeper Aleksandr Gutor.
BATE then equalised through Bressan, who ran on to Filipp Rudik's pass to slot home.
"Both teams showed some anxiety at the beginning, it was the first match after all," said BATE coach Viktor Goncharenko.
"Plzen players shook it off earlier. Our opponents were more aggressive at the beginning.
"We managed to equalise and I think that after all the draw is fair. We are satisfied given that we were behind."
