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.