Griffiths volleyed the ball on goal in the 17th minute, but Cierzniak was equal to it.
Celtic broke the deadlock three minutes later as central defender Denayer looped a header into the net after Anthony Stokes's free kick found him unmarked inside the penalty area.
Griffiths then threatened twice from long range but failed to hit the target.
Nadir Ciftci, returning from suspension, finally got Dundee United's first shot on target, but his long-range effort was easily caught by Craig Gordon.
James Forrest was looking to score his second goal in three days against United. He beat John Rankin, who was playing out of position at left-back, and took aim at the United net but Cierzniak scrambled to palm the ball onto the crossbar.
With the momentum against them, Celtic doubled their lead in the 57th minute when Griffiths volleyed into the back of the net after Scott Brown's cross.
Calum Butcher had a chance from long range in the 77th minute, but it flew straight into Gordon's arms.
Kris Commons, who scored in Sunday's Scottish League Cup final, put the United defence to the sword once again with 10 minutes remaining, finishing off a good attacking move.
However, the match threatened to end on a sour note for Celtic when Stokes was sent off for elbowing McGowan.
McGowan was dismissed himself just two minutes later for catching Liam Henderson with a dangerous tackle, the sixth red card in the three meetings between these two sides over the last 10 days.
Virgil van Dijk inflicted more misery on United in stoppage time as he capped off the scoring from a set piece.
Celtic will play Inverness Caledonian Thistle in the semi-final next month.