What flight controls should be used to counter asymmetric yaw?
The details will depend on the exact aircraft type, so I will answer for the Bombardier Dash-8 Q400 as an example.

The autopilot does not have rudder authority on the Dash-8 (source: FCOM Autoflight chapter):


  12.3.7 Autopilot
  
  The Autopilot (AP) couples the FD commands to the flight control surfaces using pitch and roll servos for automatic control of the aircraft flight path.


If not disengaged, the autopilot would try to maintain heading with roll only.

The yaw damper would not be enough to overcome the yaw asymmetry either:


  12.3.9 Yaw Damper
  
  The Yaw Damper provides through control of the rudder:
  
  
  Damping of the aircraft Dutch Roll mode.
  Turn coordination.
  
  
  The Yaw Damper authority is limited to a maximum of ± 4.5 degrees of rudder by mechanical stops. Yaw Damper commands are limited in software as a function of airspeed.


The correct procedure in case of an engine failure would be to hold rudder towards the live engine and then trim the rudder whenever time permits:

(image source: FCOM Flight Controls chapter)


  
  RUDDER TRIM INDICATOR  
  
  
  indicates trimmed rudder position
  
  RUDDER TRIM KNOB (rotary action spring loaded to neutral)
  ROTATE
  
  
  trims rudder in desired direction  
  first graduation trims slow, second graduation trims fast
  
  


The target for the rudder trim should be coordinated flight, i.e. no sideslip indicated on the primary flight display (PFD):

(image source: FCOM Indicating and Recording Systems chapter)


  
  SLIP/SKID INDICATOR (white)
  
  
  the Slip/Skid Indicator shows the lateral acceleration of the aeroplane and is a trapezoid symbol. It turns with the roll pointer
  the maximum deflection indication shows a 0.14 g lateral acceleration. This is equivalent to one and a half the thickness of the upper part of the slip/skid symbol (trapezoid shape)
  the slip skid Indicator is shown to the left when the aeroplane is slipping to the right and to the right when the aeroplane is slipping to the left
  the indicator goes out of view when the aeroplane roll is more than 60 degrees or the related AHRU attitude has malfunctioned