Roma have just one win from their six league matches so far and a growing injury list is not aiding Mourinho's cause, particularly in defence.
The 30-year-old will be reunited with Roma manager Jose Mourinho, who worked with him at Chelsea and Manchester United.
No financial details were announced but Italian media said Ligue 1 club Rennes will pay a total of around three million euros ($3.27 million) for the 35-year-old.
The defeat meant Roma failed to qualify for next season's Champions League and prompted questions about whether the 60-year-old would remain in the Italian capital, having been linked with Paris St Germain.
After an unspectacular match ended 1-1 after extra time, Sevilla ruthlessly punished the Italians in the shootout, with Gonzalo Montiel firing home the winning spot kick, just as he did for Argentina in the World Cup final against France.
"I don't think he can play. Honestly, I hope he can make it for the bench. It's the last game of the season even if after that we have Spezia," Mourinho told reporters ahead of next Wednesday's final in Budapest.
Roma youth product Bove stabbed in the winner on the rebound in the 63rd minute for his first European goal.
Portuguese manager Jose Mourinho was full of praise for his team after reaching his second European semi-final in a row with Roma, following last season's Conference League title.
Authorities reviewed security camera footage to identify a fan wearing a Lazio shirt with the name 'Hitlerson' and the number 88, a reference to the 'Heil Hitler' salute, during their 1-0 win over AS Roma.
Substitute Dybala put Roma ahead in the 64th minute, the Argentine sending a powerful shot from a tight angle inside the far post after replacing midfielder Lorenzo Pellegrini at halftime.
Jose Mourinho called on his strikers to shake off the "playmaker bug" and focus on scoring goals
Salah is in a fantastic run of form in the league, leading the charts of goals and assists in the competition with 24 league goal contributions.
The two-time Champions League winner's arrival came like a bolt from the blue for Roma's trophy-starved supporters.
Mourinho began work at Roma on Friday promising to restore long-term prosperity after the Serie A club came 29 points behind champions Inter Milan last season.
The Premier League side will play Villarreal in the final on May 26 in Gdansk.
Bruno Fernandes and Edinson Cavani each scored twice as Manchester United took a huge stride towards the Europa League final.
United will face AS Roma for a place in the May 26 final in Gdansk after the Italian side ground out a 3-2 aggregate win over Ajax Amsterdam.
Amadou Diawara struck late to earn AS Roma a 2-1 win at Fiorentina on Wednesday that keeps them in the hunt for a top-four finish in Serie A.
Italy’s top flight soccer league Serie A aims to raise at least 1.15 billion euros ($1.4 billion) per season over the next three years.
Kane netted his 200th goal for Tottenham Hotspur as they won 3-1 at Ludogorets and AC Milan’s 24-match unbeaten run