Dayton Moore tapped his hand against the back of a bench. Seated next to him, Ned Yost kicked his foot up against a nearby railing. Together they sat inside the visitors’ dugout Tuesday at U.S. Cellular Field and watched the latest edition of the Royals take batting practice.

Moore built this roster. Yost guides it. Shortly before Moore flew into Chicago to visit his free-falling club, he delivered a vote of confidence for his manager, despite the team’s blend of lethargy and ineffectiveness during the second half, and insisted Yost’s job was not in danger.

“All of our success in this organization is tied together,” Moore told The Star in a telephone conversation. “We’ve fought through some challenging times in the past. We’ll do it again. Together.”

Here in 2014, a season ticketed for October during the halcyon days of spring, the failures outnumber the successes, even after Tuesday night’s result. With Moore on hand, the Royals, 49-50, picked up a 7-1 victory over the White Sox.

Mike Moustakas boomed the team’s first home run since July 10 with a second-inning solo shot, and added his 12th of the season with a two-run blast in the eighth. Billy Butler and Raul Ibanez smacked doubles that sparked a four-run sixth inning. Bruce Chen strung together five useful frames of his own, and tied Mariano Rivera’s record for most victories ever by a Panamanian at 82.

The combination allowed the Royals to celebrate for the first time since the All-Star break. The odds against them are still long. One productive inning against White Sox swingman Scott Carroll cannot cure all of their offensive ills. But the night represented progress for a team attempting to claw back into the playoff hunt.

After being swept in Boston, one team official estimated the Royals required 40 victories in the final 65 games to make a legitimate run. One down, 39 to go.

“We need to win every game,” Moustakas said. “Every game that we play is an important game.”

As the losses piled up, the Royals exhausted options from the playbook of desperation. They called their second closed-door meeting in three days on Tuesday afternoon.