How does an aircraft maintain level flight while accelerating?
Simply put, in the scenario stated, to maintain level flight the appropriate amount of nose down input would be applied.  

As the op points out, the increase in speed through the air increases lift, and the AOA (angle of attack) needs to be reduced.  This is normally done with stick (or yoke) input, followed by trim.

On most aircraft the lowering of the nose would be accomplished by lowering the elevator (and/or raising the trim tab) on the rear horizontal surfaces.