At what point did the Democrats win control of U.S. House of Representatives during Reconstruction? Happened after the civil war

The answer to this is that the Democrats did not take
control of the House of Representatives until the Reconstruction period was very nearly
over.  They only took control in the 44th Congress -- this was the one that ran from
1875 to 1877 at the end of the second term of Pres. Ulysses S. Grant.  This means that
the Democrats took the House in the 1874 elections, largely because of a backlash
against corruption in the Grant Administration.

It was by
taking control of the House that the Democrats were able to be in position to help end
Reconstruction.  When the 1876 presidential election between Hayes and Tilden went to
the House to be settled, the Democrats were in control.  Because of this, they were able
to force an end to Reconstruction.