Joakim Soria pitched the ninth, giving up one hit, and notched his 29th save.
All of which set up the possibility of a rare accomplishment.
The Royals are on the verge of their first three-game sweep at Oakland in 20 years, since June 14-16, 1988. Two decades ago.
"That's a crazy stat right there," said Ron Mahay, who set up Soria's save with a scoreless eighth. "That'd be a good milestone for us then."
Not only that, but the Royals have not won four straight games in more than a month, or since a six-game winning streak ended on June 28.
Buck's home run proved to be the margin of victory, turning a 2-1 lead into a 5-1 advantage.
Jose Guillen walked to start the seventh, and Mitch Maier ran for him. After an out, Alex Gordon doubled. Maier stopped at third but was thrown out at the plate on Mark Teahen's roller to first.
Buck, though, smashed an 0-1 pitch from left-hander Greg Smith over the wall in left-center.
"It was a cutter. It just didn't cut," Buck said. "It was pretty straight, middle in."
Buck also made the defensive play of the game -- charging a long way and entering the Royals' dugout to catch Kurt Suzuki's foul pop to end the first inning.
"I could hear everybody saying, 'Out, out, out,'" Buck said, meaning the bench thought the ball was out of play. "But I saw the dugout and I thought, 'It's not out, it's coming back.'"
And it did.
"One of the best catches I've seen from a catcher, full-speed running," Royals manager Trey Hillman said. "No fear for his body, no fear for the step, just a great play."
Aviles connected twice for long drives off Smith. He led off the fourth with a triple that eluded center fielder Carlos Gonzalez.
"For a second there, I thought it was a sure gapper, and Gonzalez came out of nowhere," Aviles said. "I know he can fly, but I didn't know he could get there. ... He almost made a heck of a play right there."
Aviles scored on Mark Grudzielanek's sacrifice fly. Then, in the sixth, Aviles smashed his sixth home run to left field for a 2-1 lead.
"I just go up there trying to hit line drives back up the middle, and if they get a ball up and I get a good part of the bat on it, it goes," Aviles said.
Does it ever.
Meche, in his 200th career start, took his cue from Zack Greinke's dominating performance in the series opener against the A's rather young lineup.
"I watched Greinke's video and just went after these guys. They haven't been around here for a while, and he just made them feel like they were under attack all the time and I just kind of did the same thing," Meche said.
It's been a change of his own attack pattern that has made the difference for Meche in the second half.
"A world of difference since he started pitching in with the fastball and cutter," Buck said. "It just opens up his breaking ball because they've got to respect him coming in. They just can't sit on his breaking ball."
Meche went seven innings to even his record at 9-9. Mahay did his inning and Soria did his thing in the ninth. His 29 saves set a record for a Mexican-born closer, exceeding the 28 by Juan Acevedo for the Detroit Tigers in 2002.
And now, in Wednesday afternoon's series finale, the Royals have a chance to do something they haven't done in 20 years: Sweep a three-game series at Oakland.
"I was a junior in high school," said Mahay, 37. "I graduated in '89."