The renamed road sits along a stretch of Elysian Park Avenue between Sunset Boulevard and Stadium Way.
The Los Angeles City Council approved the name change and voted Friday to give its final approval.
Mayor Eric Garcetti, Councilman Gil Cedillo, Dodger CEO Stan Kasten and Scully himself spoke during Monday's ceremony, which took place at the Gate A entrance to Dodger Stadium off Sunset Boulevard.
Scully stepped up to the podium and said he felt "overwhelmed" about the entire event.
He recalled growing up in New York as a "street kid," playing baseball with a stick and a tennis ball.
What will he miss the most when he leaves the job?
"The roar of the crowd. That's what I'm going to miss the most," he said. "I don't know you, and I miss you, believe me, each and every one of you."
As a child, Scully remembered getting goosebumps when he heard the roar of college football fans on his family stereo. This sparked an interest in broadcasting, he said.
"When you folks are at the ballpark and something happens and you let out a roar, I sit there in the booth, and I don't say a word," Scully said. "For during that 40 seconds whatever, I'm 8 years old again."
Scully told the crowd that he was surprised they attended the event. The crowd responded by chanting, "Vinny! Vinny! One more year! One more year!"
He laughed in response but politely said this season will still be his last.
"I've done enough. I've said almost everything," he said in closing. "Maybe on the final day of my final broadcast, I'll somehow come up with the magic words that you deserve. As for now, I have only two magic words: Thank you."
Scully began broadcasting Dodgers games in 1950, when the team played in Brooklyn. The Bronx native said in August that the 2016 season will be his last behind the microphone.
Scully's many honors include the Ford C. Frick Award, presented annually by the Baseball Hall of Fame to a broadcaster for "major contributions to baseball" and being named the greatest sportscaster by the American Sportscasters Association.
The 88-year-old has been an announcer longer than anyone else in sports history.
City News Service contributed to this report.