Shaqiri shines as Bayern brush aside BATE
The 21-year-old Swiss international's performance was the high point for Bayern who fielded a mainly second-string team after making seven changes to the side which drew 1-1 with Borussia Dortmund in the Bundesliga on Saturday.
Mario Gomez, starting a match in the competition for the first time this season after injury, Thomas Mueller and David Alaba scored the others as Bayern, already qualified, topped Group F and avenged their 3-1 defeat away to the Belarus side.
Yegor Filipenko scored an 89th-minute consolation for the visitors who finished third and qualified for the Europa League.
Both teams finished with 10 men. Bayern's Jerome Boateng was shown a straight red card for a dangerous studs-up challenge on Artem Kontsevoi in the 50th minute and BATE's Denis Polyakov was given a second yellow for diving in the 69th.
Shaqiri, one of Swiss football's brightest prospects, was signed in the close season amid worries in his homeland that he could struggle for a place in the team behind fellow wingers Arjen Robben and Franck Ribery.
He had previously managed one goal for Bayern against lower league opposition in the German Cup.
"I think we showed today that all the players in the squad are good enough to be fielded from the start," the Kosovo-born player told German television. "I'm happy I was able to repay the trust the coach showed in me."
Shaqiri even impressed notoriously hard-to-please former Bayern Munich and Germany goalkeeper Oliver Kahn, now a television pundit.
"He also looks as if he wants to take his chance when he plays and he has the ability to dribble and unsettle opponents, which is very important nowadays," Kahn said.
Shaqiri made a lively start and set up Bayern's opener after 22 minutes as he charged to the byline and sent over a low cross which was turned in by Gomez.
BATE were denied an equaliser before halftime when Bayern defender Diego Contento forced Polyakov's effort wide from Kontsevoi's cross.
Bayern, furious about Boateng's red card, still managed to score their next two goals with a one-man disadvantage.
Bastian Schweinsteiger headed on a Toni Kroos free kick and Mueller side-footed into the net from close range in the 53rd minute for his third goal in the group.
Thirteen minutes later, Gomez returned the compliment to Shaqiri by sending over a looping cross from the left which the diminutive forward managed to head in despite the attentions of a defender who towered above him
Substitute Alaba added a fourth with a shot on the turn in the 83rd minute, the 20-year-old Austrian's first Champions League goal.