Aron Johannsson put the visitors ahead on 17 minutes but two goals in the space of three minutes early in the second half from Mitchell te Vrede and Lex Immers turned the match in Feyenoord's favour.
Ronald Koeman's side's lead lasted just six minutes, though, as Johannsson scored his second from the penalty spot in the 57th minute, but AZ were unable to extend their winning streak to four matches and reclaim top spot from Vitesse, who beat Utrecht on Saturday.
Siem de Jong opened the scoring in the 26th minute and the advantage was doubled two minutes later by Nicolai Boilesen's first for the club.
Jurgen Locadia's goal on the stroke of half-time sent the visitors into the break ahead, but a double from from Kees Kwakman in the 67th and 80th minutes turned the game around.
PSV's tally of one point from their last four matches represents their worst Eredivisie run since April 2007.
The hosts went in front through Guyon Fernandez four minutes after the break, before Twente hit back through Quincy Promes in the 72nd minute to earn a share of the spoils.
