Wilshere, who is pushing for more regular playing time in the Premier League and hoping to catch Gareth Southgate's eye ahead of next summer's World Cup, delighted the Emirates Stadium crowd with a man-of-the-match performance in the No 10 role
His powerful finish shortly before half-time came after Mathieu Debuchy and Theo Walcott had given Arsenal a commanding lead, and the Gunners, who had already clinched their place in the last-32 as Group H winners, continued in the same vein after the break
BATE defender Denis Polyakov turned a Walcott cross into his own net for the fourth, with Olivier Giroud scoring from the penalty spot before Mohamed Elneny rounded off the scoring with a fine curler
Arsenal, who fielded a totally changed team from Saturday's 3-1 defeat to Manchester United, were the dominant side from the start, with Debuchy opening the scoring with a sweetly-struck drive which was still rising as it flew into the net in the 11th minute
Walcott was denied from close-range by BATE goalkeeper Denis Scherbitsky as Arsenal continued to pour forward, but he didn't have to wait long for his goal, slotting home a calm finish after Danny Welbeck overran the ball in the BATE box
Walcott had another effort cleared off the line from Wilshere's cut-back shortly afterwards, and the pair combined again for Arsenal's third goal two minutes before the break
 This time, their roles were reversed, with Wilshere drilling a diagonal shot past Scherbitsky from Walcott's pass
It was no less than Wilshere's performance deserved, and he was back at the heart of things in the second half, sliding a fine through ball to Walcott, whose low cross was powered past his own goalkeeper by BATE's Polyakov
The over-worked Scherbitsky made a fine save from Welbeck shortly after that, but Arsenal got their fifth when Walcott was hauled down for a penalty
 Giroud was made to re-take his initial effort but he made no mistake the second time
Arsene Wenger threw on young trio Reiss Nelson, Eddie Nketiah and Joe Willock in the closing stages, but it was Elneny who capped an entertaining night for the home crowd, sending a stunning effort into the net from Wilshere's pass
There were plenty of impressive performances from Arsenal but none more so than Wilshere's
 The 25-year-old has shown signs of slowly recapturing his best form this season but this was surely his best offering yet
The goal, his first since he scored in a 4-1 thrashing of West Brom in May 2015, was the standout moment but he was excellent throughout
 His driving runs drew warm applause from the Emirates crowd and so too did his passing
 Wenger might have little choice but to start him in the Premier League soon, where he might yet hope to catch the eye of England boss Southgate
"Jack is moving forward
 You could see tonight that he is ready now to play
 He is sharp
 He is quick
 He is incisive with his dribble
 Technically he was very good tonight
 I would say for 70 minutes of the game he was dominant in midfield on the creative side and that's very good news for us
"The Gunners return to Premier League action against Southampton at St Mary's Stadium on Sunday
 They then travel to West Ham for a London derby on Wednesday
 Some of Arsenal's fringe players will hope they have put themselves in contention to start
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