How are people able to wake themselves up after a pre-specified amount of time? Auto-suggestion seems to work. Several people I know seem to be able to tell themselves "I'll wake up after 'N' hours/minutes", and do so within a minute or two of that duration. 

I thought for a moment that it might be that the duration defined is typical to the person, but even defining a weird figure (e.g., 23 minutes), it appears to work just as well as that person's regular duration of 15 minutes. 

Personally it works for me sometimes (usually when I'm under pressure), other times it does not.

How does the brain/body know to power-up after N minutes?
There is quite a lot of research on self-awakening (see this search on Google Scholar for self awakening). Hopefully someone else more familiar with this literature can add a more authoritative answer about the mechanisms of self-awakening. In the interem I briefly extract some relevant points from  Ikeda and Hashi (2012). The study does not directly address biological mechanisms. However, understanding the reliability of self-awakening and the correlates of the ability to self-awaken is presumably relevant to forming such an understanding.

First they summarise some existing research:


  Both the accuracy and success rate of self-awakening have been
  experimentally examined. More than half of the people who have the
  ability to self-awaken suc- cessfully awakened within 30 minutes of
  the predetermined time. For example, seven participants succeeded on
  nine of 14 days (64%) in a sleep laboratory, and 15 participants
  succeeded on 35 of 44 nights (80%) at their homes. Survey studies
  indicate that many people habitually self-awaken in daily life; for
  example, 52% of 269 adults (aged 21−84 years) and 10.3% of 643
  university students6 reported habitu- ally self-awakening. People who
  have a habit of self-awakening in the morning have regular sleep/wake
  schedules, tended to have a morningness chronotypology, awakened
  comfortably in the morning, and had less daytime dozing.


In their own study they found:


  The present study investigated self-awakening, both habitual and
  inconsistent, compared to awakening by external means in relation to
  sleep/wake schedules for five consecutive years in 362 students
  (starting at mean age 15.1 ± 0.3 years). Students who self-awakened
  consistently for five consecutive years (5% of all students) went to
  bed earlier than those who inconsistently self-awakened (mixed group,
  40%) or consistently used forced awakening by external means (56%).
  Awakening during sleep was more frequent and sleep was lighter in the
  consistently self-awakened group than in the mixed and consistently
  forced-awakened groups. However, daytime dozing was less frequent and
  comfort immediately after awakening was greater for the consistently
  self-awakened group than for the mixed and consistently
  forced-awakened groups.


Reference


Ikeda, H., &amp; Hayashi, M. (2012). Longitudinal study of self-awakening and sleep/wake habits in adolescents. Nature, 4, 103-109. PDF
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Matsuura N, Hayashi M, Hori T. Comparison of sleep/wake habits of university students with or without a habit of self-awakening. Psychiatry Clin Neurosci. 2002;56(3):223–224.