The visitors were ahead on 29 minutes when Karim El Ahmadi scored a well-taken volley from 12 yards after some quick passing around the edge of the penalty area.
El Ahmadi's goal proved enough to see Giovanni van Bronckhorst's men edged back in front of PSV Eindhoven, 4-0 winners at Zwolle, ahead of the champions on goal difference.
A stoppage-time goal from Jari Oosterwijk gave Twente a remarkable 4-3 victory at Groningen, where Enes Unal had scored a hat-trick only for the home side to level.
Turkey forward Unal, the 19-year-old on loan from Manchester City, gave Twente the lead on 23 minutes with a fine individual effort curled into the top-right corner from the edge of the penalty area.
Unal completed his treble with a quick-fire brace just before half-time, taking the ball down on his chest and volleying in from close range after 36 minutes and then on 39 minutes the teenager cracked home a fierce 25-yard drive.
Despite seemingly being out of the contest, Groningen, who were also pointless from their first two Eredivisie matches, fought back as midfielder Mimoun Mahi reduced the deficit on 52 minutes when he clipped the ball in through a crowded six-yard box.
Oussama Idrissi fired in from a tight angle at the far post on 69 minutes to make it 3-2 before Danny Hoesen's free-kick looked to have earned the home side a share of the points with six minutes to go.
However, with two minutes of stoppage-time played, Twente broke down the right and the ball dropped to Oosterwijk, who stabbed it under goalkeeper Sergio Padt to send the away supporters at the Euroborg into wild celebration.
Sparta Rotterdam moved up into sixth place as they beat fellow promoted side Go Ahead Eagles 1-0 with a penalty from Loris Brogno on 85 minutes.
Craig Goodwin started for Sparta but was replaced in the 69th minute, with fellow Aussie Kenny Dougall an unused substitute.
AZ completed a good week with a 2-1 victory at Utrecht.
John van den Brom had seen his men secure a comfortable 3-0 away win over FK Vojvodina in their UEFA Europa League tie in midweek.
AZ were ahead on 25 minutes through an angled drive from midfielder Marcus Henriksen after cutting in from the left, before Patrick Joosten (53) had the home side level early in the second half with a long-range effort.
With just six minutes left, Alireza Jahanbakhsh won it when he sent a low shot into the bottom left corner from the edge of the penalty area to give AZ a first Eredivisie win of the new season.