They will miss out on finishing in the top four for the first time in more than 20 years if they win their remaining matches but Liverpool and Manchester City do not drop points.
The Chilean bagged his second nine minutes later to kill off any hopes of a Sunderland comeback and seal all three points for Arsenal, closing the gap to four placed Liverpool to just one point.
Arsenal need one of their rivals to slip up to have a shot at Champions League qualification, but Wenger is pleased by the desire his team have showed in recent weeks, with the win at Stoke marking their third consecutive league victory.
"I made the decision to start Alexis at 4pm", Wenger said at a news conference.
"After that you deal with what happens".
No French Open wild card for snubbed Sharapova
Bouchard said she had "a lot of players coming up to me privately, wishing me good luck", without naming who those players were.
Arsenal now sit fifth, three points adrift of Manchester City in fourth and four behind third-placed Liverpool, although retain a game in hand on the Reds.
With only two games left in the 2016-17 Premier League season, Arsenal's chances of a Top Four finish are dwindling down to nothing!
In with the crowdWith the Premier League season coming to an end on Sunday, the Emirates Stadium looked empty at times, with the club claiming that the official attendance was 59,510.
If the Gunners beat Everton at home, and Liverpool lose or draw against Middlesbrough, Wenger's boys will scrape into the Champions League yet again.
Arsenal are hot and are playing possibly their best football of the season.
Mooy stars but Huddersfield can't break Wednesday
Keith Curles Carlisle have trailed in all three games between the sides this season but are yet to be beaten by the Devon club. We were a bit unlucky with some decisions from the referee or we were not clinical enough in front of the goal.
"Man United, they will bounce back as well next season", Wenger said.
A point on the final day against Watford away from home would be enough to definitely make sure that City finish above Arsenal, thereby securing a Top Four place.
In marked contrast to Chelsea, Arsenal have not won the league since 2004 and Wenger has come under renewed pressure this season with his side in danger of failing to qualify for the Champions League.
"Usually you lose the next game because everybody switches off, so I told the players who wanted to have a breather, "Now you can become immortal".
"I have some problems", he added.
BBC presenter touches a Jeremy Corbyn supporter's BREAST
Brown was mid-interview with assistant political editor Norman Smith on the BBC News Channel when the passer-by stepped into shot. Not seeming to appreciate the presenter touching her body, the unknown female slaps him on the shoulder then walks off.
The away side have been at the bottom for most of the season, losing 24 of their 36 matches, winning just six.