Twente gave the ball away and Wright-Phillips exchanged passes with Jo before running into the penalty area and firing into the bottom corner.
Robinho forced a good save from visiting goalkeeper Sander Boschker soon after as the home side looked to press home their advantage.
City were looking comfortable with their lead but that only lasted as long as the 17th minute when they paid the price for a lapse in concentration.
The home defence hesitated, waiting for the referee to stop play when Elia seemed to catch Pablo Zabaleta as they went for a bouncing ball, but the whistle never came and Elia ran on to the loose ball and planted a low shot past Joe Hart.
Twente, with McClaren cutting an animated figure on the touchline, were lifted by their equaliser and were the better side for the rest of the half.
Indeed, the visitors could have had a second goal had Javier Garrido not denied Blaise Nkufo with a last-ditch tackle as the striker shaped to shoot inside the box.
City, in contrast, went backwards but they did spark into life before the break when the otherwise ineffective Jo danced past Douglas in the area but saw his weak shot easily saved.
The home side were much improved at the start of the second half and Wright-Phillips had already gone close when Robinho struck on 57 minutes.
There seemed little danger when the Brazilian picked up the ball on the left-hand edge of the area but he showed his class by smashing a superb curling shot past the helpless Boschker.
The game looked up for Twente when Benjani, making his first appearance of the season after a thigh injury, added a third five minutes later with a shot that came off Douglas.
But the pressure was quickly back on City when Wielaert was allowed to rise unmarked at the near post and head home from a corner.
Twente kept coming forward and Huysegems' first chance to level came and went on 70 minutes when he scuffed his shot wide when unmarked on the edge of the area.
City were still threatening too and Robinho went agonisingly close to sealing victory - twice firing against the same post after running at the Dutch side's defence.
But McClaren's men will feel they did enough for a point and should have earned one right at the end when the ball fell to Huysegems inside the six-yard box but he skied his shot hopelessly over the bar.
Manchester City boss Mark Hughes:
"Robinho could have had a hat-trick. But we were grateful for the strike for the second goal. He has fantastic ability in those areas.
"He is one of the best players I have seen in that position to be able to manipulate the ball and create things inside the box.
"Overall, we are delighted. We have got a positive result and that is what we need to achieve at home to progress."
FC Twente boss Steve McClaren:
"We felt we should have got something from the game. We certainly had chances.
"But the character of the players was magnificent after we went a goal down so early against City's front five.
"We dug in, and it was end to end. It was an open game, which must have been great for the neutral - but for both managers it was edge-of-the-seat stuff."
Man City: Hart, Zabaleta, Richards, Dunne, Garrido, Wright-Phillips, Ireland, Fernandes, Vassell (Elano 66), Robinho, Jo (Mwaruwari 59).
Subs Not Used: Schmeichel, Onuoha, Ben-Haim, Hamann, Evans.
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