Tang, 29, will replace Chu, whom Lee announced earlier this month he would appoint as the city's assessor-recorder, a position left vacant after Phil Ting was elected to the state Assembly in November.
District 4 includes the Sunset and Parkside neighborhoods on the western side of the city.
Chu and Tang will be sworn in to their new posts on Wednesday, Lee said.
Tang, who has served for more than five years as Chu's legislative aide, will provide continuity in the district, the mayor said.
"She truly represents this district and all its diverse views," he said.
Tang, who came to today's news conference with her parents Tony and Jih-Ying Tang, has lived in the Sunset District since she was a year old and graduated from Lowell High School. She then graduated from University of California at Davis in three years.
Tang said her long history in the district will influence her work as supervisor.
"It's difficult to separate my work life and personal life," she said, citing concerns about parks, San Francisco Municipal Railway service and other neighborhood issues that she has grown up with.
Tang said she found out on Monday evening that she was being appointed when Lee called her as she was driving home.
"He told me I should probably stop driving before he gave me the news," she said.
Tang is the same age that Chu was when she was initially appointed to the supervisor seat in 2007 to replace former Supervisor Ed Jew, who was suspended and later convicted of state and federal charges including mail fraud and bribery.
Considered one of the more moderate members of the board, Chu then won re-election to the seat and has chaired the budget and finance committee during her tenure as supervisor.
"She set a great example of what a supervisor should be," Tang said. "I have big heels to fill."
Board president David Chiu said Tang will be received with open arms by the other supervisors, most of whom have extensive experience working with her in Chu's office.
"Everyone likes her, respects her, and knows her to be incredibly hard working and deeply knowledgeable about city government," Chiu said.
Tang will face an election this November to try to retain the seat.
The mayor's first appointee to the board, Christina Olague in District 5 to replace newly sworn-in Sheriff Ross Mirkarimi in 2012, did not win re-election last November.
Olague made several controversial votes during her short term, including to reinstate Mirkarimi, who was suspended by Lee after being convicted in a domestic violence case involving his wife.