The Brazilian repaid another chunk of his £32.5million British record transfer fee with a sublime strike in the 57th minute when the scores were level at 1-1.
Stephen Ireland fed Robinho just inside the area and the diminutive forward side-stepped his marker before letting fly with a powerful right-foot shot which nestled into the very top corner of the goal.
Five minutes later it was 3-1 and effectively game over for former England manager Steve McClaren's new side when substitute Benjani Mwaruwari found himself clean through and his low shot took a deflection as it wrong-footed the keeper.
Twente captain Rob Wieleart was left unmarked at a corner and reduced the arrears with a looping header just three minutes later, but the Dutch side were unable to claw their way back into the game for a third time despite playing with plenty of heart.
Before the second-half drama, the first period had been an intriguing battle as City grabbed an early lead only to hand the initiative back to the visitors.
City boss Mark Hughes had named an offensive-looking starting line-up with fit-again Darius Vassell joining Robino, Jo and Shaun-Wright Phillips in a powerful attacking quartet.
Captain Richard Dunne passed a late fitness test on a knee injury, while Javier Garrido and Gelson Fernandes also returned to bolster a side that was beaten at Bolton Wanderers on Sunday.
Wright-Phillips took less than three minutes to break the deadlock as the lively England winger played a neat one-two on the edge of the box with Jo before drilling the return pass low beyond the keeper with a composed finish from 12 yards.
However, Twente were handed a lifeline midway through the half when a lapse in the home defence gifted Eljero Elia an equaliser.
The City defence stood motionless anticipating a free-kick after a stooping Pablo Zabaleta took a blow to the head but, as the referee waved play on, Elia skipped through the middle and planted a low shot past Joe Hart.
Hughes cut a furious figure in the home dugout and his mood darkened as the Dutch side gained the upper hand in a keenly-contested first half.
Jo could have lifted the tension when he beat his marker in the box only to see his low shot repelled by the legs of the keeper five minutes before the interval.
Robinho then set up Jo with an angled run and lay-off early in the second half but the wasteful Brazilian was muscled off the ball and his stabbed effort rolled harmlessly to the keeper.
Robinho then showed his compatriot how to finish with his 57th minute strike, before the two sides traded another goal each and City played out time in relatively comfortable fashion.
Robinho found time to strike the post twice in the final 15 minutes with angled shots which beat the keeper and on another night he could have been celebrating his second hat-trick in City colours.
Twente substitute Stijn Huysegems fired over the bar from six yards in the final minutes to leave McClaren holding his head in his hands but the City fans celebrating the perfect start to their European group campaign.