The Scottish Cup quarter-final tie between the two sides at Tannadice Park ended in a 1-1 draw, meaning that the teams will now contest a replay.
The game was a hugely controversial one and had no fewer than three players sent off - two for United and one for Celtic.
Virgil van Dijk of Celtic and United's Paul Paton were sent off early in the game, and Terrors defender Paul Dixon was dismissed later for blocking a goalbound shot.
Griffiths himself failed to convert a penalty and both teams hit the post, as the game ended in a 1-1 draw.
Following the end of the match, Scotland international Griffiths told BBC Sport: "The sending offs were a bit harsh but that was the referee's call. The penalty looked soft.
"After Paul Dixon was sent off it was one-way traffic. I was disappointed to miss the penalty but I tried to put the header in the corner and thankfully it went in.
"We went in 1-0 down and were very disappointed with that. I'm just delighted I scored and kept the team in the Cup.”
Nadir Ciftci put hosts United in the lead from the penalty spot on the stroke of half time, and Griffiths restored parity for the Scottish Premiership leaders on 71 minutes.
United and Celtic will now play a replay at Celtic Park, and the winners of the tie will take on Inverness Caledonian Thistle or Raith Rovers in the semi-finals of the Scottish Cup.