"It's just kind of mind-boggling to know my name is next to his," Jeter said after the Yankees' 4-2 victory.
Jeter had a chance to break the record in the eighth inning, but he walked against reliever Grant Balfour.
The Yankees are off Thursday, and Jeter will get his next opportunity Friday night at home against Baltimore.
Already on their feet in anticipation, fans at Yankee Stadium let loose with a roar when Jeter's sharp grounder inside the first-base line got by a diving Chris Richard in the seventh.
Jeter's parents, watching from an upstairs box between home plate and first base, raised their arms and exclaimed in excitement.
The Yankees trailed 2-0 at the time, and Jeter said he reluctant to acknowledge the mark because he didn't want to "disrespect" the Rays.
Rays players and coaches clapped as Jeter stood at first base. He took off his helmet and twice waved it to the crowd of 45,848 during an ovation that lasted about 2 minutes.
"What an ovation I got from the fans," he said. "I've been trying to do it for them."
After entering the game in an 0-for-12 slump, his longest hitless stretch this season, Jeter broke out of the rut with a bunt single toward third base leading off the bottom of the first inning. He beat the play without a throw, bringing a standing ovation from the crowd.
"That's why I bunted in the first inning. I needed to get one hit, right?" he said.
With cameras flashing all around the ballpark on every pitch to Jeter, he grounded out in the third inning against Rays starter Jeff Niemann and drove a ground-rule double to straightaway center in the fifth.
On his first chance to tie Gehrig, Jeter came through in fitting fashion - with an opposite-field hit.
In the middle of the eighth inning, the large video board in center field showed a replay and flashed "Congratulations Derek!" Jeter also swiped second base in the first inning for his 300th career steal, which ranks second on the franchise list behind Rickey Henderson (326).
Gehrig's final hit came on April 29, 1939, a single against the Washington Senators. The Iron Horse had held the club record for hits since Sept. 6, 1937, when he passed Babe Ruth.