The Daggers played against 10 men for 87 minutes after Gulls defender Aaron Downes was sent off for a foul on Abu Ogogo, Medy Elito netting the resulting penalty.
Joss Labadie equalised before Josh Scott put the hosts ahead again in the 85th minute and the Daggers then looked set for victory.
But after Luke Wilkinson fouled Lloyd Macklin, Rene Howe held his nerve under pressure to slot home an 89th-minute spot-kick.
"I'm disappointed we didn't win, you'd expect us to when they went down to 10 men," Burnett said. "But it's not as easy as that because the opposition can get bodies behind the ball and make it difficult for you.
"However, we were ahead and we needed that second goal to kill off the game. We didn't get it and that meant they were always in the match, even though they only had 10 men.
"We played well but they played in the second half and had some good chances. It's just a shame we couldn't kill off the game and get the three points."