Roma had won its opening two Serie A games without conceding but fell behind to a Dusan Vlahovic free-kick inside 76 seconds at Allianz Stadium on Sunday (AEST).
Juve had a second goal through Manuel Locatelli ruled out for a Vlahovic handball in the build-up in a half the home side dominated, outshooting its opponents 10 to three.
Mourinho responded by replacing Gianluca Mancini and Leonardo Spinazzola with Stephan El Shaarawy and Nicola Zalewski at the break, and Roma improved in the second half.
The visitors found a way through from one of their three efforts on target in the second period, with Tammy Abraham turning in former Juve player Paulo Dybala's acrobatic pass.
👍🏻 @PauDybala_JR proved his worth for @OfficialASRoma at his old stomping ground, providing this assist for @tammyabraham as @juventusfc let its lead slip in the big one in #SerieA. All the highlights are here 📺 https://t.co/FGnBSLp8DG | #JuventusRoma pic.twitter.com/ewUmsHdEzX— beIN SPORTS (@beINSPORTS_AUS) August 27, 2022
While his side's unbeaten start to the season remains intact with a credible point in Turin, however, Mourinho accepted Roma was rather fortunate.
Asked what he said at half-time, Mourinho said: "I told the team I was ashamed of them. I felt ashamed of being their coach.
"The game changed, but let's not talk about tactics; let's instead talk about attitude. We can't get where we want with this attitude.
"I told [assistant coach Salvatore] Foti to pray that it finished only 1-0. That would have been a good result after that first-half performance.
"I told them to take advantage of the luck we had, knowing the game could have been over at that point. Every now and then, you are dominated but you have to manage it better.
"I had a bench with very few offensive solutions, especially compared to [Massimiliano] Allegri. But after analysing the second half, we deserved to win that 1-0."
The goal Roma conceded came from Vlahovic's first touch and was Juve's first direct free-kick goal in the league since Cristiano Ronaldo scored against Torino in July 2020.
It ended the Giallorossi's four-game run without conceding in all competitions, but the home team were unable to hold on and now have just one win from three games this term.
Juve head coach Allegri recognised his side could have no complaints with the full-time scoreline after it failed to finish off Roma when on top in the first half.
"Jose's always smart and sharp in the way he reads games," Allegri said of his opposite number. "If you don't kill off the game, you always leave the possibility of an equaliser.
"We were tired after a strong first half and probably should have focused on passing the ball around to slow things down.
"That's something we need to learn. You cannot expect a team to dominate the match for the full 90 minutes."
Juve, which had beaten Roma in 10 of their 11 previous league games in Turin, went with a 4-2-3-1 formation, but Allegri hinted he is open to change once at full strength.
"First of all we need to focus on getting Federico Chiesa, Paul Pogba and Angel Di Maria back," Allegri said when asked about his set-up.
"They all have technique and a change of pace. Having changes available from the bench makes a big difference."