Under what authority can the AIM overrule the FARs? A commonly held belief among pilots is that the FARs are regulatory, while the AIM is not. Violation of a regulation in 14 CFR is punishable by the FAA, while the AIM contains mostly "good ideas" and violation of its precepts is only punishable to the extent that it rises to the level of FAR 91.13 (careless and reckless operation). The FARs must be published in the Federal Register and generally be subject to the rulemaking process, while the AIM can be updated by the FAA at any time, with no input from anyone outside the Administration.

With that background, consider a note in the AIM, at the end of section 5-4-5(l) about Fly Visual approach segments, on page 5-4-20 (page 346 of the current PDF version of the AIM):


  NOTE−
  The  FAA  Administrator  retains  the  authority  to  approve
  instrument approach procedures where the pilot may not
  necessarily have one of the visual references specified in
  14 CFR 
  §
   91.175 and related rules. It is not a function of
  procedure 
  design to ensure compliance with 
  §
  91.175. The
  annotation  “Fly  Visual  to  Airport”  provides  relief  from
  §
  91.175 requirements that the pilot have distinctly visible
  and identifiable visual references prior to descent below
  MDA/DA.


This is, as far as I can tell, the only location where this exemption from FAR 91.175 is described. FAR 91.175 is quite specific in the elements needed before continuing beyond the DA/MDA, and does not stipulate that the Administrator has the authority to authorize exemptions. For an example of one that does, consider FAR 91.117(a): 


  Unless otherwise authorized by the Administrator, no person may operate an aircraft below 10,000 feet MSL at an indicated airspeed of more than 250 knots (288 m.p.h.).


What authority gives the AIM the ability to carve out an exception to the FARs? Are there other situations where the FARs are contradicted by a non-regulatory document without explicitly allowing exemptions or authorization by the Administrator?
The issue you have appears to be that the AIM describes a case where procedures don't follow a rule (91.175) and you don't see where the authorization comes from for that deviation.  

What you're missing is that the approach procedures are in fact rules themselves.  They just don't get published in Title 14 CFR.  Standard instrument procedures are covered by Part 97, Subpart C.  It essentially states that the procedures are controlled via the FAA Order 8260 series.  

So if the approach procedure specifies minima or other procedures outside the norm of a separate rule, the procedure is the controlling rule.