Earlier in the tie, Shapovalov had won his match with a 6-4, 6-3 scoreline against Pablo Carreno Busta.
Bautista Agut kept piling up the pressure, as he finally emerged victorious following a high-quality two hours and 44 minutes of tennis.
Felix Auger-Aliassime upset World No.3 Alexander Zverev to carry Canada through to the ATP Cup semi-finals on Thursday.
The reigning champion will play the winner of Group C in the semi-finals on Saturday.
It was Hurkacz, who sealed the tie for Poland with a convincing 6-1, 6-4 win against Diego Schwartzman.
The World No.3 has started 2022 the same way he finished last, earning ATP Cup victories against Cameron Norrie and Taylor Fritz for Team Germany.
Pablo Carreno Busta overcame Serbia’s Filip Krajinovic 6-3, 6-4 to book Spain’s place in the semi-finals at ATP Cup in Sydney.
Poland defeated Argentina to become the first country to qualify for the semifinals of ATP Cup
Alexander Zverev defeated Amrican Taylor Fritz 6-4, 6-4 to keep alive Germany’s ATP Cup semi-final hopes on Tuesday.
Despite losing the match, he was satisfied with his development and was happy to participate in the match.