The Gunners' penchant for lapses in concentration struck again however and after a ball from Aleksei Rios was whipped in Mirko Ivanic was on hand to thump his header past David Ospina at the near post.
BATE pulled another back to push Arsenal all the way but in the end it was a comfortable enough victory for Wenger's side. His faith in his squad depth was repaid in style, as Arsenal picked up a valuable three points in their bid to escape their Europa League group. Wenger made a decision to go with Schalke's Sead Kolasinac, a relative unknown for many fans outside of Germany.
One of the main talking points of the first half came when Joseph Willock, on a rare start for Arsenal in central midfield, appeared to be slapped in the face by Nemanja Milunovic.
Three minutes into the second half, Giroud had his landmark goal and Arsenal their fourth when what looked like the softest of penalty awards for a challenge by Stanislav Dragun on Shkodran Mustafi gave him the chance to roll home from the penalty spot.
Nelson's skill and overall display drew particular praise from Arsenal legend Martin Keown, who lauded the teenager on BT Sport by comparing him to Neymar no less.
In their absences the senior players include Per Mertesacker, Jack Wilshere, Olivier Giroud and Theo Walcott, meaning the promising Ainsley Maitland-Niles is among those likely to feature, and the manager said: "This is the kind of job: you do not get 10 opportunities to play for Arsenal".
Wenger threw down the gauntlet to Arsenal's young players before the game. Arsenal got stuck into their sausages and ended up having the better of a rollicking, bundles of chances, blast in Belarus.
He'll have been less pleased with the costly slip before BATE's second goal. The 30-year-old led the line effectively at the Borisov Arena, linking play with his back to goal and allowing Walcott and Wilshere to run in behind.
Theo Walcott scored the opening two goals in the ninth and 22nd minute.
Everton rallied to 2-1 ahead and looked set to rebound from its previous 3-0 loss to Atalanta. Nicklas Bendtner, from the penalty spot, Anders Konradsen and Vegar Eggen Hedenstad put the hosts firmly in control before Juan Felipe's last-minute consolation goal.