Roma led 1-0 from the first leg against Vittesse but fell behind in the 62nd minute at the Stadio Olympico before Tammy Abraham prevented extra-time with a stoppage time equaliser to ensure qualification to the final eight, where they will once again face group stage opponents Bodo/Glimt.
Roma and Lazio meet in a pivotal Serie A clash on Sunday, with European qualification up for grabs.
The two teams are separated by just one point, with Roma in seventh place and Lazio in fifth.
The ever controversial Mourinho has made reference to Sarri's chain-smoking habit ahead of the encounter and has rued the fact that while his players were exerting maximum energy to ensure qualification, his opponents were conserving energy for the derby della capitale.
"Obviously, I am not happy that we played today with many players who will also have to be there on Sunday while tonight Lazio are home smoking cigarettes with Sarri, but it’s because Vitesse are really good," Mourinho told reporters after the game.
"Now Sarri is at home smoking his cigarette while I go home trying to think how we can recover from this for Sunday."
He added: "We never came into this to control the result or aim for a draw to get us through. We worked with one objective, preparing to go for victory.
"It was a bad performance on a technical level, it might seem tactical, but it was technical. We defended well and showed character, but we also pushed hard at the end with El Shaarawy as a third striker, not a wing-back.
"I don’t want to say we 100 per cent deserved to go through, but over the two legs did a little more."
