Michel Jansen's side fell behind just before half-time but showed their character with two goals in the space of 10 second-half minutes at the Parkstad Limburg Stadion to take all three points and leapfrog Vitesse and champions Ajax.
Roda went ahead in the 39th minute as Guus Hupperts rose at the back post to meet an Ard van Peppen delivery and powered his header past Nick Marsman for his second league goal of the season.
Twente slowly worked their back into the game, though, and levelled through Luc Castaignos with 21 minutes remaining.
The former Inter striker received a pass from Youness Mokhtar before slotting home his eighth goal of the season.
And Twente's comeback was completed soon after as Quincy Promes struck the decisive goal - his seventh of the season - to seal their second straight win after last week's 5-2 thrashing of NAC Breda.
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