Price comes from behind to advance at PDC World Championships
Woodhouse won the opening set, averaging just under 105 (104.95) compared to Price's 93.39.
The lead was short-lived, though, with the 2021 world champion claiming the next three sets to win his opening match of this year's tournament.
The second set saw went all the way down to a fifth and final leg, with Price winning the decider before going on to take the next two sets and advance to the third round.
He will face either Raymond van Barneveld of the Netherlands or Englishman Ryan Meikle in the next stage of the competition.
Price is one of a record-breaking six Welsh players at the competition and will be looking to add to the three PDC World Championship titles he has already gathered across his career.
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