After dominating a goalless first half at De Grolsch Veste, Twente eventually took the lead 10 minutes into the second period when Luc Castaignos slotted the ball beyond Vitesse goalkeeper Piet Velthuizen.
The visitors' chances of mounting a comeback were dealt a blow five minutes later when Patrick van Aanholt was sent off for a late challenge on Dusan Tadic.
And the in-form Serbian topped an impressive display by scoring his side's second goal with a 78th-minute lob - his 10th goal in the league this season.
The result sees Twente climb to within a point of league leaders Ajax, who play on Sunday, while Vitesse could be eight points behind the Amsterdam club come the end of the weekend.
Den Haag's resurgence following the departure of Maurice Steijn continued as they climbed out of the automatic relegation spot with a 1-0 win at Roda, who went bottom.
Mitchell Schet scored the only goal of the game in the 53rd minute, as Den Haag went 16th.
It could have been an even better day for Den Haag, had NEC not recorded an impressive 3-1 win at RKC Waalwijk to climb to the safety of 15th position.
Romeo Castelen put the hosts in the ascendancy when he scored in the sixth minute at the Mandemakers Stadion.
However, NEC responded well and went in at the break 2-1 up thanks to goals from Navarone Foor and Lasse Nielsen.
English striker Michael Higdon made sure of the win midway through the second half to take NEC to within two points of their opponents in the table.
In the day's remaining clash, Utrecht gained a valuable point in their own fight against relegation with a 1-1 draw at AZ Alkmaar.
