Cabrera went 1-for-4 with four putouts in his first game, as the Yankees defeated the Indians, 7-2.
"He may have been pressing a little at the plate," said Torre. "But the most important thing for us is his play in center field."
"I feel really happy that I got my first hit here at Yankee Stadium, and I'm happy to have played my first game," Cabrera said after the game. "I felt no pressure. It was fun playing in front of 52,000 fans."
Cabrera, 20, opened the season with Double-A Trenton, hitting .267 with nine home runs and 44 RBIs in 75 games. He was promoted to Triple-A Columbus on June 28, where he hit .324 with three homers and 11 RBIs in nine games.
"I wasn't looking for this. It was a surprise," said Cabrera through Rivera. "I'll do my best to help [the Yankees] here. I want to do work at everything so I can get better and better."
To make room for Cabrera on the 25-man roster, the Yankees placed right-handed pitcher Carl Pavano on the 15-day disabled list with right shoulder tendinitis, retroactive to June 28.
"I've been preaching that most of our answers are going to have to come from within," said general manager Brian Cashman. "We have to fix ourselves with what we have, and this is part of that process. We're not afraid to go with some youth."
Cabrera's defensive ability is the primary reason for his callup, as the Yankees have been hurt several times by subpar defense in center field. Cabrera, who was ranked as the organization's No. 8 prospect last year by Baseball America, was listed as the system's best defensive outfielder.
"We've heard nothing but good things," Torre said. "We want him to just go out there and have some fun, play the outfield the way [he knows] how."
Several center fielders have been mentioned in potential trades to New York, including Mark Kotsay and Mike Cameron, but the Yanks will give Cabrera a try to see if they can solve the problem from within.
"I've had a lot of discussions with opposing clubs about trades, and center field is a thin market," Cashman said. "The clubs that have one, the cost is astronomical."
With Hideki Matsui back in his familiar position, left field, the Yankees have lacked both defense and production in center, where Bernie Williams and Tony Womack have been platooning.
Williams is hitting .259 with five home runs and 35 RBIs, but it has become apparent that he is no longer capable of producing on an everyday basis.
Womack, who was moved to the outfield from second base in early May, when Robinson Cano was promoted from Columbus, is hitting .243 with no homers, 12 RBIs and just five extra-base hits in 71 games this season.
Torre plans on starting Cabrera in center field for the entire four-game series against Cleveland, then taking the All-Star break to analyze what he sees.
"They told me that I'm here to play," said Cabrera, who wasn't ready to name himself as Williams' successor. "Time will control that. Right now, I'm just here to play."
Cabrera becomes the third rookie to play a significant role with the Yankees this season, joining Cano and Chien-Ming Wang.
"All the time I've been here, with all the criticism the farm system has taken, it's refreshing," Torre said. "The philosophy is the same: We want to win now. I think in light of players that you're looking at, the consensus tells us to try this."