MPs have supported both Houses of Parliament moving out during the multibillion-pound restoration programme.
They committed to a "full and timely decant" after deleting sections from a motion which would have allowed them to kick into the long grass the need for "comprehensive works" at the Unesco World Heritage Site.
A body would also be established to provide up-to-date costings of the work and a "realistic" timetable for the repairs.
The amendment was tabled by Public Accounts Committee chairwoman Meg Hillier and was viewed as a way of ensuring more rapid progress to repair the Palace of Westminster compared with two motions tabled by Commons Leader Andrea Leadsom.
But Conservative former minister Sir Edward Leigh warned during the debate that Ms Hillier's amendment could result in MPs leaving Parliament for up to 12 years and effectively authorise the spending of £5 billion during a time of "unparalleled austerity".
Ms Hillier's amendment was supported by 236 votes to 220 - majority 16.
The amended motion was then voted upon and approved by 234 votes to 185, majority 49.
Peers are to consider the restoration and renewal of the palace on February 6.