How did reanimated Haku and Zabuza die? According to Naruto wiki, a reanimated body can die from any of the following ways:


  
  Remove the soul from the reincarnated body, such as with the Dead
  Demon Consuming Seal.
  Have the summoner end the technique. As they are unlikely to do this
  voluntarily, using a genjutsu to trick them into cancelling the
  technique is ideal. 
  A less reliable, though equally effective method is to have the
  reincarnated soul be affected in some emotional way that gives them
  closure, allowing their soul to break free of the Impure World
  Reincarnation's influence.
  


At episode 265, after Haku and Zabuza got reincarnated, they were faced by Kakashi and his army. At first, they were really hesitating to kill Kakashi and army and said that were under control of someone. Meanwhile, Kabuto used Mitarashi Anko to harness Orochimaru's chakra to tighten Haku's and Zabuza's soul to the body and as a result they became really powerful, a immortal killing puppet. They were fighting really hard and at episode 266(spoilers below):


   Both Zabuza and Haku got killed by Kakashi's chidori. 


How did they die? Neither of the above three points were followed i.e the soul were not removed by reaper death seal, nor the summoner was under a genjutsu nor the soul was affected in any emotional way. Yes, they were emotional at first but after getting Orochimaru's power, their soul were tightened more to the body which Kabuto felt that they were now a mere immortal killing machine.


   Why did Zabuza and Haku got killed by a mere ninjustsu?
Neither Haku or Zabuza were "killed" in the sense you describe. In the same episode you referenced, they were both temporarily immobilized: 


Haku was immobilized by shielding himself in front of Zabuza when attacking was attempting to strike him with Chidori
Zabuza was immobilized after Kakashi successfully hit him with a Chodori.


As they were immobilized, this gave Ensui Nara an opportunity to hold them with the Shadow Bind technique


  After Zabuza cuts through Haku, Ensui told Santa that Kakashi only caught one enemy and immobilised Haku as he regenerated. When Kakashi successfully connected his shadow to Zabuza's, he immediately immobilised him as well.


In this Youtube video, you can see the shadow line holding two shinobi down. This gave another shinobi to use the Cloth Binding Technique to capture them.