We describe in detail an atom-by-atom exchange manipulation technique using a scanning tunneling microscope probe. As-deposited Mn adatoms (Mn(ad)) are exchanged one-by-one with surface In atoms (In(su)) to create a Mn surface-substitutional (Mn(In)) and an exchanged In adatom (In(ad)) by an electron tunneling induced reaction Mn(ad) + In(su) --> Mn(In) + In(ad) on the InAs(110) surface. In combination with density-functional theory and high resolution scanning tunneling microscopy imaging, we have identified the reaction pathway for the Mn and In atom exchange.