Pique headed in Andres Iniesta's "incredible" cross to give Czech goalkeeper Petr Cech no chance after he had made several fine saves to put his side within sight of an unlikely draw.
Cech reacted well to deny Alvaro Morata and David Silva in the first half while Czech defender Roman Hubnik deflected Morata's shot against his own post.
Spain were given a fright when a stretching Hubnik forced a save from David de Gea and Cesc Fabregas made a goalline clearance to keep out Theodor Gebre Selassie's header.
But Pique's late intervention ensured the champions began with a victory to join Croatia on three points, a result which delighted King Felipe VI who congratulated the players after the match.
"We struggled a bit during the first half and we'd already had a lot of chances," Pique told journalists.
"In the second half, we were lucky to score," he added, before paying tribute to his brilliant team mate with the observation: "Iniesta's pass was incredible."
Czech Republic started aggressively, but Cech was kept busy during the first half, setting the tone of the match with stops from Morata and Silva, who was making his 100th appearance.
Spain's rhythm improved as Czech intensity dipped, and while coach Pavel Vrba's ploy of employing two full backs on the right flank to deal with Jordi Alba was initially successful, the left back began to find space.
Both Alba and Silva forced stops from Cech, while Tomas Necid struck a shot at De Gea before half time.
"I think we played well," Pique continued. "We controlled the match but the ball just didn't want to go in."
"We took the initiative throughout the match," added Spain coach Vicente del Bosque. "We dominated."
The Czechs swapped their aggressive approach for more disciplined containment after the break, although Morata's cross forced Hubnik into a deflected error which beat Cech but bounced back off the post.
Vrba's side also served warning that they were capable of extending their opponents' miserable run of failing to win any of their opening tournament matches since 2008 with Hubnik and Gebre Selassie going close.
Ladislav Krejci was on hand to make a superb last-ditch tackle on substitute Thiago Alcantara as a Czech defence that conceded 14 during the qualifiers kept Spain's wealth of attacking talent at bay.
Del Bosque introduced Aritz Aduriz for Morata, but it was an unlikely candidate, Pique, who ultimately secured the winning start for the champions.
(Reporting by Martyn Herman, editing by Adrian Warner and Ian Chadband)
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