Draper had been the last Briton standing in the singles after Cameron Norrie and Dan Evans lost on Monday.
He was the better of the two in the second set, taking advantage of an increasingly faltering opponent to break to love for a 3-2 lead, and closed the set on a wild Hurkacz forehand.
The refocused Pole, who won his first clay-court title at the Estoril Open on Saturday, broke quickly in the third but could not serve out the match at the first attempt as Draper upped his return game.
However, Hurkacz was the stronger in the resulting tie-break to set up a meeting with Spain’s Roberto Bautista Agut.
Earlier, third seed Carlos Alcaraz withdrew from the tournament with a muscle injury in his right forearm.
The Wimbledon champion had been due to play Canada’s Felix Auger-Aliassime on Wednesday and practised in Monte Carlo with his forearm taped.
Alcaraz is the defending champion at the Barcelona Open, which begins on 13 April.