Having lost 2-1 to PEC Zwolle on the opening day of the season, Feyenoord have suffered their worst start to a season for 50 years and half of the six goals they have leaked have been scored by former players.
On Sunday they were reduced to nine men as they lost their first home league match since November 2011 -- a 28-match run.
Jordy van Deelen was dismissed 12 minutes into the second half conceding a penalty that Dusan Tadic converted to put Twente 2-1 ahead after Kyle Ebecilio, a former Feyenoord youth player, had equalised Graziano Pelle's opener for the home side in the first half.
Tadic made it 3-1, also from the penalty spot, after 73 minutes, by which time the hosts were playing with nine men after Stefan de Vrij received a second yellow card.
It went from bad to worse for Ronald Koeman's side when Luc Castaignos, another Feyenoord old boy, made it 4-1 in the 79thminute.
"This man isn't pro-Feyenoord. That's my opinion after twoyears as Feyenoord manager," Koeman said of referee Richard Liesveld.
Reigning champions Ajax did no better, losing 3-2 at AZ Alkmaar despite twice leading through goals by Christian Eriksenand Siem de Jong.
Viktor Elm scored Alkmaar's winner after 74 minutes.
Ajax ended with 10 men after Ricardo van Rhijn was sent off in the 65th minute.
Groningen beat Utrecht 2-0 to join three other clubs on six points at the top of the standings.