The 33-year-old will be tasked with keeping the O's in the Football League after they dropped into the League Two relegation places on Saturday following a 2-0 defeat at Mansfield Town.
Webb, who was appointed Orient first-team coach this summer, became assistant manager when Edwards was appointed on November 23 and has now been given a second promotion in quick succession.
Edwards confirmed earlier on Sunday that he was leaving the club after just two months in the hotseat.
"It is with regret that I have made the decision to stand down as manager of Leyton Orient FC," Edwards said in a statement.
"I have been offered and duly accepted a role working within the coaching department of the Football Association.
"I thank the chairman for the opportunity to manage the team and I thank all staff, players and supporters for the backing that they have given to me in my time here - I wish Leyton Orient every success in the challenge ahead and in the longer-term future."
Webb's first game in charge will be the visit of promotion-chasing Carlisle United to the Matchroom Stadium this weekend.