"What exactly motivated this rivalry in the first place is not totally clear to me," Pires told the Daily Mail. "But I believe towards the end it was all to do with Manchester United's excessive arrogance and their excessive physical play.
"With each triumph, (United manager Sir Alex) Ferguson would insist that his team was the best and that never went down well at Arsenal. These things irritate players.
"But United is extra special. It is another step up from the other two. At the start of my time, United had been the traditional champions for a decade and believed they held some kind of power over London clubs.
Arsene Wenger and Ferguson have had daggers drawn for years, which has been reflected on the pitch, with each manager blaming the other for a series of incidents that have required FA intervention.
Pires added: "In the dressing room before a United game, Wenger would have one message every time.
"He would just ask us to be calm and to play with our usual style and not to be affected by the opposition.
"Specifically, he would tell us not to enter into wars with people like Roy Keane, Wayne Rooney or the Neville brothers.
"He would tell us that he knew what United would do but that we had to be professional. But Wenger badly wanted us to beat them.
"In the run up to those games he would remind us of the rivalry and tell us what it meant to our supporters.
"There was no extra money on offer, of course. The win bonus would be the same for United as it would, for example, against a team like West Brom. But Arsenal players don't need an extra incentive when they face United. It's all in their heads already."