Spurs had won just two of their last seven away games in European competition before Friday (AEDT) and their struggles on the road showed in a timid opening 45-minute showing.
However, the visitors were unfortunate to not go ahead after the interval at the GelreDome as Bryan Gil rifled onto the woodwork in an otherwise uneventful first hour of football.
Nuno Espirito Santo's side, which rested Harry Kane entirely for the trip to Holland, was eventually made to pay for its lacklustre performance when Wittek drilled in from the edge of the area to end Spurs' unbeaten start in the competition and leave them third in Group G.
Pierluigi Gollini smothered the first chance of the game as he raced out to deny Nikolai Baden Frederiksen before Jacob Rasmussen flicked narrowly wide following Wittek's corner.
Nuno's side could not manage a shot on target in the first half but went in level at the break after Thomas Letsch's team failed to make the most of their control of proceedings.
Spurs duly returned after the interval with greater attacking impetus, Bryan rattling the crossbar with a left-footed curler just two minutes into the second half.
Eli Dasa then drove narrowly wide of the right-hand post before forcing another save from Gollini as Vitesse piled on the pressure just past the hour mark in a contest of limited opportunities.
The host had scored in each of their last eight games and made it nine when Wittek rifled a left-footed half-volley - via a deflection off Davinson Sanchez - into the bottom-right corner with 13 minutes remaining.
Spurs pressed on for a late equaliser that did not come, meaning their five-game unbeaten run in the Netherlands in the 21st century came to an end.