The latest slip-up from Steve McClaren's men meant Twente dropped to fifth in the Dutch league on 44 points with Vitesse Arnhem, 5-3 winners at Heracles Almelo thanks to Wilfried Bony's treble, moving into third on 45.
Rajiv van la Parra opened the scoring for Heerenveen after 15 minutes but nine minutes after the break Dmitry Bulykin brought Twente level with a close-range header.
Finnbogason, however, restored the lead for Marco van Basten's Heerenveen 11 minutes from time before a late second booking for Twente's Edson Braafheid.
"This has nothing to do with our coach and he has all the support of the squad," said Twente midfielder Leroy Fer.
"This is now the same story for six matches already that we fall behind too easily and that has to change."
Bony and Marco van Ginkel handed Vitesse an early 2-0 lead at Almelo, who replied before the break through Lerin Duarte.
Early in the second half, Remco Pasveer saved a penalty from Bony but from the following corner Van Ginkel headed home his second goal.
Ivorian Bony added two more goals to seal his hat-trick and the win for Vitesse while lifting his season's total to 22.
Elsewhere, Michael de Leeuw scored a stoppage-time winner for Groningen to sink PEC Zwolle 1-0 at home, while strugglers VVV Venlo beat RKC Waalwijk 1-0 thanks to Marcel Seip's goal.
