Simon got off to a confident start, breaking early in the first set before comfortably taking it 6-1.
The second set went with serve until Udomchoke forced a break late on, eventually sealing the set 6-3.
The third set was again even-handed, with the action going with serve to force a tie-break.
Thailand's Udomchoke got the better of the 12th seed to take the third set and pave the way for a potential upset.
While the 30-year-old battled for a surprise victory, Simon managed to take control of the match and force through a fifth and final set in Melbourne after claiming 6-3 in the fourth.
The Frenchman continued his dominance for the closing stages of the clash after a tough fight from his Thai competitor and booked a spot in the second round of the tournament.