Medvedev was defaulted from his semi-final against Novak Djokovic in Beijing for hitting a spectator with a ball and ended the 500-level event with zero ranking points and no prize money.
The former world number one said it was an accident.
Playing five days later in Shanghai, sixth-ranked Medvedev overcame an early wobble to hold off 59th-ranked Struff.
He sent down 21 aces to advance.
"When I broke him I felt like I could win it," the 30-year-old Medvedev said of a dominant third set in which he despatched a masterful backhand volley during the sixth game that helped him go up 4-2.
The court was "different from Beijing", he told an on-court interviewer, adding "it was a bit tough to get used to".
He faces either Czech qualifier Dalibor Svrcina or Tomas Martin Etcheverry of Argentina in the next round.