A spokesman said the band didn't disclose which finger Page hurt or how the injury happened.
Page is expected to start rehearsing again in three weeks, which would put the iconic rock band on track to perform Dec. 10 at the O2 arena. The concert to honor late Atlantic Records founder Ahmet Ertegun was scheduled for Nov. 26.
"Led Zeppelin have always set very high standards for ourselves," Page said, adding that the delay would "enable my injury to properly heal" and permit the band "to perform at the level that both the band and our fans have always been accustomed to."
Tickets for the original show will be honored on the new date and ticketholders who cannot attend on Dec. 10 will have until Nov. 14 to request a refund, the BBC said.
The last time Led Zeppelin performed a full concert together was in 1980.
More than 1 million fans registered to buy 20,000 tickets, sold by lottery, for their latest concert.