Axelrod, one of six rookies on the Sox's 12-man pitching staff, limited the Yankees to six hits in seven innings. He kept the Sox close enough for Dayan Viciedo to launch a three-run homer off David Robertson in the ninth inning to vault the Sox to a 4-3 victory at warm Yankee Stadium.
Rookie Addison Reed retired Derek Jeter on a deep fly to right and the tying run at first to end the game with his 11th save.
"I love it, especially against the Yankees," Reed said. "That's my first time in this stadium, and it was an awesome feeling. I love those situations."
The Sox overcame Mark Texeira's home run in the eighth that pushed their deficit to 3-1. Rookie left-hander Hector Santiago had struck out Alex Rodriguez and Robinson Cano before allowing the homer.
"They're going to learn under fire and survive," Robin Ventura said.
Viciedo, 23, narrowed his focus after Rapada's error.
"I was thinking fastball down the middle," Viciedo said.
The Yankee Stadium environment and the star-studded lineup he faced certainly didn't overwhelm Axelrod.
"I wasn't that excited out of the ordinary," he said.
Axelrod and rookie left-hander Jose Quintana, who will face his former organization Friday night, comprise two-fifths of the Sox's rotation. And the promotion of left-hander Leyson Septimo gives the Sox four rookie relievers as they try to maintain their 2 1/2-game lead over the Indians in the American League Central while facing the likes of the Yankees, Rangers and Blue Jays before the All-Star break.
"You can't get experience until you're given an opportunity," veteran left-hander Matt Thornton said. "A lot of these guys have taken this opportunity and are running with it. They've done a great job."
Axelrod's only blemish was in the fifth when he allowed three consecutive hits with two outs that led to two runs.
"Against a team like this, you have to mix it up," Axelrod said. "I think A.J. (Pierzynski) did a great job (calling the game). We were on the same page for the most part. I did a lot of preparation, watching video with (Jake) Peavy."
Sox pitching coach Don Cooper relishes the task of holding a first-place lead with youth and said that commitment was a testament to the organization for developing and finding young talent.
"I'm telling you. I'm having a blast with this," Cooper said. "We're embracing the challenge, and that's what we want them to do — embrace this opportunity."