An Arsenal line-up featuring a number of fringe players made light work of its Belarusian opponent, taking the lead after 11 minutes when Mathieu Debuchy ended a three-year goal drought with an accurate finish.
Theo Walcott, who dynamism and pace caused BATE problems all evening, scored Arsenal's second from close range before Wilshere buried its third from the edge of the penalty area before half-time.
Walcott forced an own-goal by Denis Polyakov shortly after half-time and Olivier Giroud added a fifth from the penalty spot before Wilshere teed up Mohamed Elneny for the sixth as Arsenal swaggered to a victory that provided an ideal tonic after a 3-1 home defeat to Manchester United in the Premier League.
Arsenal went into the game having already secured first place in Group H, but the emphatic nature of the victory and a strong performance by Wilshere will have pleased Wenger, while BATE's European dream is over after it finished bottom of the group with one win from six games.
It took Elneny just two minutes to latch onto Giroud's knock-down and lash a shot towards goal from just inside the penalty area but Denis Scherbitski dived to his left and parried it away.
Arsenal's patient probing paid off nine minutes later when Walcott held the ball up in the penalty area and Debuchy stepped up to power a shot into the top-left corner of the net with his left foot, leaving Scherbitski no chance.
The BATE goalkeeper had to make an acrobatic one-handed save to prevent an own-goal by Polyakov and moments later he saved at Walcott's feet after an unselfish header from Giroud put the winger through at close range.
Wenger's men dictated the game with crisp passes and elegant build-up play before Danny Welbeck's trickery took him into the box after 37 minutes and, just as it looked as though the ball had run away from Arsenal, Walcott pounced and fired it home from the edge of the six-yard box.
Walcott turned provider six minutes later when he knocked the ball to Wilshere on the edge of the box and the former first-team regular steadied himself before hooking the ball into the right corner of the net with his left foot.
Wilshere threaded a throughball for Walcott to chase on the right wing after six minutes of the second half and the England winger's cross would have found Welbeck had the on-rushing Polyakov not slammed it into the net in an attempt to clear the danger.
Shortly after the hour mark, Walcott was brought down in the box by Nemanja Milunovic's lazy tackle and Giroud, who was ordered to take the penalty twice do to an encroachment, plundered the spot-kick into the bottom-right corner of the net with his trusty left foot.
Wilshere turned provider in the 74th minute, collecting the ball inside the box before calmly squaring it to Elneny, who curled the ball into the far corner of the net with an exquisite right-footed shot.
Though BATE had a brief reprise in the final 10 minutes, Arsenal's defence, marshalled by Rob Holding and Calum Chambers, held firm.