The Gunners were 2-0 down at the break following a lacklustre first-half display but were much improved after the restart and earned a point thanks to late goals from Aaron Ramsey and Olivier Giroud.
The tie with Besiktas is delicately poised after a 0-0 draw in Turkey, which saw Ramsey sent off, but Wilshere says the comeback at Goodison Park will give Arsenal huge belief they can progress to the group phase.
"This was a fantastic point and now we have a big game on Wednesday and this will give us great confidence going into that," Wilshere said, speaking to Arsenal Player.
"We have got great character, great togetherness and some leaders out there that keep us going. We also have great players to go with that.
"We never give up and at this level that is massive. We have always shown over the years that Arsenal are going to battle like mad right until the end.
"There have been a few games where we have been behind right until the end but have kept going."
Giroud is a doubt after picking up an ankle injury in the dying minutes against Everton and Ramsey is suspended for the game following his dismissal last week.
Wenger will be able to call upon his Germany internationals Per Mertesacker, Mesut Ozil and Lukas Podolski who were unavailable for the first leg after they were afforded extra recovery time following their exploits at the World Cup.
Arsenal’s game against Besiktas will be shown live on Sky Sports 5HD, with coverage beginning at 7.00pm.