Shakhtar struggled in their opening two group games, letting slip of a two-goal lead in the final four minutes of their previous match against Porto to draw 2-2.
But there was little danger of that happening in Borisov on Tuesday as they blitzed BATE in the opening 45 minutes and became the first team to score six times in the first half of a Champions League game.
Luiz Adriano plundered a 12-minute hat-trick, the third fastest in Champions League history, adding a fourth just before the break, after Alex Teixeira had opened the scoring in the 11th minute.
Douglas Costa also netted, seven minutes after Luiz Adriano got his first from the penalty spot before the latter added another from the spot late on.
The striker is only the second player to score five times in a Champions League game after Barcelona's Lionel Messi.
Luiz Adriano's haul means Shakhtar coach Mircea Lucescu will now be relishing the return game in two weeks as he eyes a place in the knockout stages.
BATE had initially appeared in control of a scrappy opening, with their visitors camped behind the ball early on.
That was until it was given away cheaply in midfield in the 11th minute and Taison skipped away to thread the perfect throughball for Teixeira to round the goalkeeper and open the scoring.
Taison almost repeated the trick at the wrong end five minutes later, as his loose backpass was picked up by Maksim Volodko, who beat goalkeeper Andriy Pyatov - only for Darijo Srna to produce an impressive diving clearance.
Shakhtar duly doubled their lead in the 28th minute when Taison and Teixeira combined in almost identical circumstances to the opening goal. However, this time goalkeeper Sergey Chernik felled Teixeira as he raced through and Luiz Adriano tucked away from the penalty spot.
BATE began to chase the game and were picked off on the break.
Shakhtar's third goal, in the 35th minute, came from a BATE corner, with Taison instrumental once again. The Brazilian took the ball into the BATE half before exchanging passes with Luiz Adriano and slipping through for Costa to finish.
The hosts had barely regained possession when yet more poor defending allowed Luiz Adriano to run across his defender and convert Vyacheslav Shevchuk's low cross from the left at the front post two minutes later.
The striker completed his hat-trick on 40 minutes as he reacted quickest as the ball dropped and defenders watched, to thrash in from close range.
And he took his tally for the night to four one minute before half-time when two BATE defenders followed the ball instead of tracking the on-rushing Shevchuk, who squared for Luiz Adriano's easiest strike of the lot.
Shakhtar looked the most likely to score after the break as Luiz Adriano hit the post with a powerful header from Bernard's cross in the 66th minute.
But his fifth did arrive from the penalty spot in the 82nd minute after being brought down in the box himself.
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