The Gunners host FA Cup winners Wigan at the Emirates Stadium on Tuesday night looking to move back above Tottenham into fourth place as the race for elite European football is set to go to the wire.
Wenger's squad have long fallen short in the quest to launch a sustained assault on the Barclays Premier League title. But the Arsenal boss insists that must change, saying: "Last year we made 70 points and finished third. This year, we can make 73 points and finish fourth, so what you want to do is have a consistency and fight for the Premier League."
The Frenchman added: "It (finishing fourth) is the minimum required. You could see that when Chelsea's Champions League position was under threat, they came out quickly and said 'no, what is most important for us is to be in the Champions League next year'.
"It is difficult to achieve, but we have done it over the years. Is it enough? No. We want more, but this year, Manchester United was above everybody else."
Wenger is set to be handed a bumper £100million summer transfer budget to strengthen the squad, with £20million-rated Montenegro striker Stevan Jovetic from Fiorentina said to be high on his shopping list.
He added: "We have rebuilt the team and since January we have certainly done very well if you look at the number of points we have taken compared to the other teams. We have created a good basis and that stability can help us start strongly next season."
With Sir Alex Ferguson retiring, Wenger will become the longest-serving manager in the country after taking charge at the Gunners in 1996, but has yet to commit his long-term future past the end of next season.
The French coach, 63, has seen rival clubs deliver more silverware under a succession of different managers.
He said: "You get sometimes that feeling, that because of the money thrown at the club, the impatience is there and the drive for success and demand for success is much higher. It is because these people want to be rewarded quickly and that could explain why the instability is higher."