Jeff Mathis caught three of the four games in Kansas City, with Bobby Wilson handling the fourth.
Conger was in the No. 8 spot in the order, between first baseman Mark Trumbo and center fielder Peter Bourjos, his former teammates at Triple-A Salt Lake.
Conger, a switch-hitter with power and discipline from both sides of the plate, made his Major League debut as a starter last September with Weaver on the mound. It turned out to be a memorable night for both.
Weaver handed Conger the game ball to go with a souvenir ball he gathered after his first big league hit, a two-run single in the first inning that helped create breathing room for the ace.
"It was his first win," Weaver said, grinning, after the 7-0 decision over the Indians in Cleveland. "It kind of caught me off guard when I came in and saw he was catching. Before the game, I knew he was nervous -- I've been around a little bit.
"I told him to go out and have some fun. I said, `If I don't like the pitch you call, I'll shake you off.' He did his homework. He watched tape of their hitters and my tape and knew what I wanted to do. And he brought it into the game. It was a great experience for him -- and me."
A two-out single by Shin-Soo Choo in the fourth inning provided the Tribe with its lone baserunner through seven innings. Weaver, having thrown 100 pitches in dispatching 21 of the 22 hitters, turned it over to the bullpen for the final six outs.
The club's 2006 first-round Draft pick, Conger went the other way to deliver a two-out, two-strike single during a four-run first inning.
"I was pretty nervous," said Conger, who spent time with fellow receivers Mike Napoli, Mathis and Bobby Wilson before the game. "Everyone's been great to help me feel comfortable. I was a little surprised [by the starting assignment]. I watched as much video as possible."