Muriel gave the visitors the lead in the 33rd minute, but Denis responded from the penalty spot for the hosts, who remain two points behind their visitors in 10th place.
There was little danger of either side winning the game in the second half as they seemed happy enough to settle for a point as the fog loomed around the Stadio Atleti Azzurri d'Italia.
Atalanta made most of the early running with Giuseppe De Luca almost picking out Carlos Carmona in the 27th minute, but Roberto Pereyra managed to intervene to clear the danger.
Udinese struck with their first real chance of the game in the 33rd minute when Muriel reached Giovanni Pasquale's cross at the far post to slot in past Andrea Consigli.
Atalanta's response was immediate with Cristian Raimondi testing Zeljko Brkic with a header, but Emmanuel Badu almost doubled Udinese's lead on the counter attack.
The home team were gifted a route back into the game in the 39th minute when a penalty was awarded when contact was made between Gabriele Angella and Federico Peluso inside the area.
Denis stepped up to bring Atalanta level before half-time.
Denis could have made it two straight after the break, but Badu got in a desperate saving tackle to deny him after De Luca's shot was parried by Brkic.
Giacomo Bonaventura and De Luca both went close with efforts before Udinese coach Francesco Guidolin decided to send on top scorer Antonio Di Natale.
The Italy striker had little chance to make an impression, though, as the two teams slowed down the pace of the game and simply cancelled each other out.
In the 25 minutes Di Natale spent on the field, there was not one dangerous shot on goal at either end and a draw was a fair reflection of the play.