The Royals are 4-2 in Interleague Play this season and 24-18 over the past three years. DeJesus has hit .339 in his career against NL clubs and Meche's career record against them is 11-4.
"We need to switch leagues," DeJesus said, laughing. "No, I'm just kidding."
DeJesus belted the first four-run homer of his career against right-hander Micah Owings, with thanks to Meche's single.
Jose Guillen led off the fourth with a double, Mark Grudzielanek singled and John Buck was hit by a pitch to load the bases. Owings got the second out and up came his rival pitcher, a 2-for-18 batsman. But Meche rapped a sharp grounder that glanced off Owings, and he beat a throw by second baseman Orlando Hudson for an RBI single.
Meche swung at a 1-0 fastball.
"It was a good pitch, and I knew it was probably the only really good pitch I was going to get in that at-bat to get Dave back up to the plate again," Meche said.
"It was a high one-hopper and he just barely tipped it and I saw it was going to be a tough play, so I took off. If I don't get on base, that [the grand slam] doesn't happen so it's a different game."
That gave DeJesus his chance and, boom, his drive soared over the right-center-field fence and almost landed in the stadium swimming pool. The Royals had a five-run inning and a 6-0 lead.
"Fastball," said DeJesus. "He threw a ball first pitch and I said, 'Let me see a fastball. If I get it, I'm going to put a good swing on it.' And I was able to get it up and on an arc."
DeJesus knocked in his fifth run, a career high, with a sixth-inning double and had three hits. In his career against the NL, he's 82-for-242, .339, with four homers and 34 RBIs.
Meche, known for his knee-buckling curveball, changed up on the Diamondbacks and attacked with his fastball.
"Teams know he has a really good breaking ball -- he's lived on it early in his career -- and now he's using it to his advantage by not throwing it," Buck said.
Meche struck out the side in the first inning and finished with 10 strikeouts in his 7 1/3 innings. He gave up just four hits and walked one.
"They're an aggressive team with a lot of young guys," he said. "Going into the game, I knew they'd be hacking and I tried to use that to my advantage."
Sure did. Meche retired the first 12 batters, seven on strikes. Conor Jackson singled to open the Arizona fifth and the D-backs picked up two runs in that inning.
"Minimum damage, it could have been worse," Royals manager Trey Hillman said. "Lot of strikeouts. We knew they could strike out coming in."
After being shut out in the 10-inning series opener, the Royals outscored the D-backs 20-6 in the past two games and whacked 31 hits. That was a huge awakening for the run-starved Royals.
"That's a big deal for us right now with what we've been through this season, some of the ups and the downs, and hopefully this is the start of a big upswing for us," Hillman said.
Every starting player in the lineup, except for Alex Gordon, had a hit on Sunday. Grudzielanek banged three singles and raised his average to .305. Guillen had two more hits and has a 10-game streak (21-for-42, .500).
Guillen boosted his RBI total to 53.
For DeJesus, the grand slam and five RBIs in a game were career firsts. Fittingly, they came on Father's Day, a salute to dad Heryk DeJesus who taught David and his brothers how to hit.
"Once I get on the bus, I'm going to give him a call and say, 'Hey, thanks for that one,'" DeJesus said.