Do Mahayana Buddhists explain birth and extinction in the same way?
There is no difference in manner or extent between the rebirth we experience from moment to moment and the rebirth we experience from life to life. To misunderstand this is to misunderstand the subtlest meaning of the Buddha's teachings on emptiness or anatta.

Therefore, think about what this means for your question. When an enlightened one first becomes enlightened and reaches nirvana does an enlightened one continue to exist from moment to moment? Does he not exist from moment to moment? Both exists and does not exist? Neither exists nor not exists? Whatever the answer... then to that exact same extent and manner an enlightened one will continue to exist from life to life. To say otherwise is to assert that something has materially changed such that after nirvana an enlightened one continues in that very life, but utterly ends or is annihilated at the end of that life. This cannot be the case. It would mean an enlightened one substantially existed or inherently existed in some way and that this ended or was annihilated at death.

It is important to know the manner in which the three poisons end. The real liberation comes through knowledge or the extinction of ignorance. From this, the other poisons are extinguished as they rely upon ignorance to sustain them. Nirvana is nothing more than waking up to the knowledge of reality as it really is rather than by perceptions and conceptions clouded by ignorance. 

Think about what Nagarjuna is saying here:


  
  It is not asserted that the Blessed One exists after his passing away; nor is it asserted that he does not exist, both exists and does
  not exist, or neither exists nor does not exist.
  Even while he is living, it is not asserted that the Blessed One exists; nor is it asserted that he does not exist, both exists and
  does not exist, or neither exists nor does not exist.
  There is no distinction whatsoever between samsara and nirvana; and there is no distinction whatsoever between nirvana and samsara.
  The limit of nirvana and the limit of samsara: one cannot even find the slightest difference between them.
  


Verse 17 and 18 assert that there is no distinction between the way an enlightened one exists from moment to moment as compared to life to life.

Verse 19 and 20 draw on the prior verses to show there is no distinction between nirvana and samsara. Nothing has materially changed except where before there was ignorance now there is knowledge.

It is like waking up from a dream even as the dream perceptions continue... there is no longer any doubt that one is perceiving a dream. Not one iota of doubt. The ignorance of the dreamer has been replaced with the sure knowledge of an awakened one.