An open but tightly fought battle in the Stadio Olimpico confirmed Group C will be a three-way battle for qualification.
An open but tightly fought battle in the Stadio Olimpico confirmed Group C will be a three-way battle for qualification after Chelsea punished Azerbaijani newcomers Qarabag 6-0.
Numerous saves by Alisson, notably in the second half when Atletico were pushing hard, frustrated Diego Simeone's side as the Argentine coach returned to a stadium he knew well as a Lazio player.
"We have to compliment the goalkeeper, as Alisson did a great job," Simeone told Italian television after the game.
"The shots were on target, aside from Saul hitting the woodwork, as otherwise he had saves on Koke, two or three on Saul – he had a wonderful performance.
"The most important thing in football is not aesthetics, but goals, and we lacked a goal, so Roma deserved their point. I hope this point can be valuable for the future so we continue in the Champions League."
Italian coach Eusebio Di Francesco conceded Roma were struggling with their fitness levels.
"We had an excellent first half. We dropped off a bit then, notably physically, but I'm satisfied the team responded well to this suffering," said Di Francesco.
"It's a positive result against a great team. They played two finals in four years while we're still under construction.
"For 50 minutes I saw excellent football on our side, with character. Of course we can improve our physical condition. I love Spanish football, it's a role model to follow."
Brazilian goalkeeper Alisson proved decisive in defying Luciano Vietto, Saul Niguez early in the second half and two Angel Correa searing efforts on 64 and 83 minutes and another Saul header in time added on.
Saul's first chance was at the start of the game, but his powerful shot grazed the post with Gregoire Defrel and Antoine Griezmann also shooting wide.
The Italian team reacted and Slovenian Jan Oblak stopped a powerful shot by Belgian Radja Nainggolan from the edge of the area after 30 minutes.
That move gave rise to Atletico's best chance of the first half -- a Vietto centre that Koke failed to finish off.
In the second half, Atletico pushed forward in front of the Roma goal.
It was then that the importance of Alisson came to the fore, as he halted a Vietto effort after a great pass from Saul in the 56th minute, and minutes later made a prodigious stretch to deny Argentine Correa.
Alisson got his hand to save another powerful shot from Correa and then on the rebound, saved a Saul header.
It has been a frustrating start to the season for Atletico -- runners-up in 2014 and 2016 to Real Madrid -- after also being held to two draws in three La Liga games.
French striker Griezmann was back after suspension but was off form, failing to finish his three best chances in the first half before being substituted late on.
Atletico host Chelsea in their next group game on September 27 with no room for error, while Roma will go in search of points and goals in Azerbaijan.
"I have a positive feeling," added Simeone.
"It's going to go down to the wire between Roma, Chelsea and us. We're improving and are on the right track but efficiency is the weapon of great teams.
"It didn't go exactly as we planned since we didn't score but for everything else it was almost perfect."