The American was pushed to his limits in the first set, with both players exchanging breaks of serve before Fish produced some punishing ground strokes to take the tie-break 7-3.
Soeda refused to crumble following the disappointment in the opener and was once again a threat on Fish's serve, breaking his opponent on two occasions.
However, Fish was urged on by the New York crowd and forced the set into another tie-break before extending his lead with a 7-3 triumph.
Fish produced his best tennis of the match in the opening stages of the third set and was able to quickly break Soeda's serve early on.
Soeda was unable to recover from the setback and Fish improved his serve to wrap up the third set 6-3 to book his passage into the next round.