Suarez went closest to breaking the deadlock when he sprinted on to a through ball to go one-on-one with Huesca's Andres Fernandez in the second half but failed to successfully dribble around the keeper.
Atletico continued to bombard the home side in the closing stages but kept being thwarted by Fernandez, who got low to deny Joao Felix in the final minute, while Huesca's defenders also did their bit to get their bodies in the way.
The draw brought Atletico back down to earth after their 6-1 win over Granada on Sunday in their first game, in which Suarez struck twice after coming off the bench, with Huesca doing a far more effective job of shutting down Diego Simeone's side.
"We gave everything but in the end we didn't create the chances we wanted to," said Atletico midfielder Saul Niguez.
"We expect to be fighting for the La Liga title this season like everyone else, it's a pity it didn't work for us today, I think we deserved to win but we're playing against a tough team who didn't let us get the three points."
Huesca midfielder Borja Garcia said his side had to sacrifice their usual style of play to get a result.
"We knew they were a very good team and would dominate play, they would press us and suffocate us and that's what they did, especially in the first half," he said.
"We normally like to have the ball and play possession football but that wasn't going to happen today, we had to focus on defending and we are happy with the point we have got considering who we were up against and how well we played."