If Villarreal coach Unai Emery was unhappy at the space his side left Barca in the opening goal, he will have been furious as Fati doubled his tally just four minutes later.
The game was over by halftime as the 18-year-old produced a magnificent display that will only give strength to those who see him as the natural successor to Leo Messi.
Olimpia must defeat Delfin at home and rely on other results going their way to have any chance of progressing to the knockout phase.
To survive without taking out new loans, the club's new board has refused grants from its former chairman Clemens Tonnies. The millionaire and meat-baron stepped down after two decades as an official following widespread fan protest.
World champion Xu Jiayu from Zhejiang clinched his second gold in the men's 100m backstroke in 52.60 seconds, after his 200m backstroke victory on Sunday.
In Tuesday's other action, Shanghai smashed Hebei 3-0 (25-18, 25-12, 25-7), Tianjin beat Yunnan 3-0 (25-12, 25-17, 25-13), Liaoning defeated Guangdong 3-0 (25-18, 25-12, 25-15), and Zhejiang crushed Sichuan 3-0 (25-12, 25-14, 25-16). Enditem
The other two German participants - RB Leipzig and Borussia Monchengladbach - have harder tasks in Groups F and B respectively.
Despite the goal, Real Madrid had to thank keeper Thibaut Courtois for the win after the Belgian made a series of impressive saves.
Unai opened the scoring with a carefully placed shot following a swift break in the 40th minute and, although Kike Garcia equalized for Eibar three minutes into the second half, the midfielder popped up again to finish the best move of the game three minutes from time to deservedly send the points back to Bilbao.
Williams injured her Achilles tendon during her semifinal loss to Victoria Azarenka at the US Open earlier this month.
Second seed Nadal will face Italy's Stefano Travaglia in the round of 32. The latter survived a five-set thriller against Japan's Nishikori Kei to advance 6-4, 2-6, 7-6 (7), 4-6, 6-2.