But that's not how it works. You didn't vote for the President and Vice-President, you voted for members of the Electoral College. In turn, they voted for the positions of President and Vice-President, each on a separate ballot. Biden's election to the Vice-Presidency is completely separate from Obama's election to the Presidency. The fact that politics dictate that the two are selected as a pair does not change the underlying Constitutional process.
Even if this is this the case, which I do not think it is, all the Cabinet is out of office immediately and has to go through re-confirmation by the Senate.