Midfielder Mowatt twice put the home side ahead with brilliant curling efforts, but Iceland international Gudmundsson responded to each of them.
A Marco Silvestri blunder give him his first leveller and he then beat the Italian from the spot after a Giuseppe Bellusci foul.
The outcome was perhaps a little harsh on Leeds who were the more positive side, but it means they are now eight without a win and that Neil Redfearn has one point from his two games since taking the head coach position.
The game did not get going until the second half, as the two were tentative in a first period which needed warming up as winter began to take its first real hold on the English season.
Souleymane Doukara headed wide for Leeds and Mirco Antenucci stung the palms of Stephen Henderson, while for Charlton George Tucudean tried a backheel from a Callum Harriott cross from the left that went wide.
Athletic had the first 45’s best opportunity in the 26th minute but it was more by accident than design as Silvestri missed Johnnie Jackson’s corner and gave Andre Bikey an open goal to aim at. He bundled it, rather than make a clean connection, and Stephen Warnock was able to hook off the line.
Leeds resumed a relative level of control after that, but the best they could muster was an Mowatt drive that skimmed over and two off-target efforts from Antenucci.
Mowatt – Leeds’ standout player in defeat to Cardiff on Saturday – found a way to pep things up after the restart, though.
After Tommaso Bianchi knocked down a Warnock cross, Charlton missed their chance to clear and the ball fell to Mowatt on the penalty spot. He shied away from shooting with his unfavoured right foot and his chance looked to have gone, but he went a yard further back and then curled a beauty into the top corner with his left.
It was a third of the season for the teenage midfielder and gave Leeds the lift they had been looking for, and Antenucci went in search of a second, having one hit charged down and then going wide with the rebound.
Leeds may have been on top but they were pulled back 62 minutes in.
They failed to clear a cross from the left and the ball fell to Gudmondsson who had little right to score from close to the byline, but he drilled a shot in and Silvestri let it under his body and it squirmed in.
But Mowatt was not willing to see his brilliance be wasted and produced another piece of magic to get his side back in front in the 65th.
After some heavy pressure the ball fell to Warnock who shaped to shoot but instead squared to Mowatt who whipped in another top-corner curler from 20 yards.
Just like after the first goal, though, Charlton hit back. They got a corner with 10 minutes to go and in a crowded box referee Graham Salisbury saw an infringement by Giuseppe Bellusci, pointed to the spot and Gudmondsson did the rest.