With Bopanna's defeat, India's challenge has ended in the elite clay-court tournament.
Six-time champion Novak Djokovic has withdrawn from this year's Italian Open, organisers said on Tuesday.
Four-times Grand Slam champion Alcaraz, who is gearing up to defend the French Open title later this month, fired off 24 winners in a strong performance to seal victory.
Backed into a corner, world number five Paolini mounted a comeback of her own in the second set, winning six successive games after herself trailing 4-0.
World number three Novak Djokovic came back from a set down to beat unseeded Swiss Stanislas Wawrinka 4-6, 6-3, 6-3 and claim the Rome Masters title on Sunday.
Federer's bid to win his first title on the clay courts of the Rome Masters juddered to a premature halt on Wednesday when he lost his second-round clash against Spain's Albert Costa 3-6, 6-3, 6-2.
Russian Maria Sharapova ground down Carla Suarez Navarro 4-6, 7-5, 6-1 to win the Rome Masters final on Sunday, the 35th WTA Tour title of her career.
Ailing Briton Andy Murray suffered another setback to his French Open preparations as he was ousted 6-7, 6-3, 6-2 by Richard Gasquet in the third round of the Rome Masters on Thursday.
Rohan Bopanna and Pablo Cuevas came from behind to knock out seventh seeds Feliciano Lopez and Marc Lopez, sealing a quarter-finals berth at the ATP Rome Masters.
Jannik Sinner held onto his world number one ranking ahead of his return from a three-month doping ban
Meanwhile, in the WTA event, third seeded-pair of Sania Mirza and Yaroslava Shvedova lost 3-6, 6-7(7) to second seeds Yung-Jan Chan and Martina Hingis in the semi-finals of the $3,076,495 red clay event.
Roger Federer settled a score with his bogey player Guillermo Canas on Tuesday, easing his way into the third round of the Rome Masters with a 6-3, 6-3 victory.
The seventh seed Indian pair of Mahesh Bhupathi and Rohan Bopanna crashed out of the ATP Rome Masters after suffering a straight-sets defeat at the hands of Marcel Granollers and Marc Lopez, in Rome.
They are yet to play a match in the men's doubles event at the Rome Masters.
Third seed Andy Roddick lost 6-0, 6-4 to last 16 opponent Juan Ignacio Chela.
Rafael Nadal enjoyed the perfect fillip ahead of his French Open defence with a commanding 7-5, 6-3 victory over world number one Novak Djokovic in the delayed Rome Masters final on Monday.
Third seeds Leander Paes and Mahesh Bhupathi crashed out of the Rome Masters after being outplayed in the super tie-breaker by Americas Sam Querrey and John Isner in the second round. The ace Indian pair was beaten 3-6, 6-3, 5-10 by their unseeded opponents in 72 minutes.
The world number one went down to Spain's Albert Costa in the second round of the Rome Masters.
Second seed Andy Roddick crashed out in the first round of the Rome Masters, losing 7-6, 6-1 to Guillermo Canas.
The Briton he lost in the third round to slow court specialist Mariano Zabaleta.
Holder Novak Djokovic took advantage of Roger Federer's slump after a rain break to win 4-6, 6-3, 6-3 on Saturday to set up a Rome Masters final against world number one Rafael Nadal. Nadal, a triple champion here, powered past Chilean 12th seed Fernando Gonzalez 6-3, 6-3.
Mahesh Bhupathi and Rohan Bopanna ended runners-up at the Rome Masters after a tame straight sets defeat at the hands of top seeded American team of Mike and Bob Bryan in Rome on Sunday.
Mahesh Bhupathi and Rohan Bopanna entered the quarter-finals of the Rome Masters, but it was curtains for Leander Paes and his Austrian partner Jurgen Melzer.
Czech Tomas Berdych snapped a run of 11 consecutive defeats by world number one Novak Djokovic to reach the semi-finals of the Rome Masters before Rafael Nadal was stretched to the limit against David Ferrer on Friday.
The third seeded Indo-Czech pair beat Romanian Andrei Pavel and Germany's Alexander Waske in the quarter-finals.
Third seeds Leander Paes and Lukas Dlouhy moved to the quarter-finals of the ATP Rome Masters after getting a walk-over from Croat Ivan Ljubicic and Michael Llodra of France in the second round.
Sania Mirza and Martina Hingis of Switzerland stormed into the semi-finals of the Rome Masters with a straight-set victory over Julia Goerges and Silvia Soler-Espinosa on Thursday.
Rohan Bopanna and Aisam-ul Haq Qureshi failed to get past the formidable United States' combination of Mike and Bob Bryan, losing in straight sets in the second round of the ATP Rome Masters, on Thursday.
Novak Djokovic said a majority of players believe favouritism is at play in the tennis anti-doping system.
World No. 1 Novak Djokovic beat Roger Federer in straight sets on Saturday and will play Rafa Nadal in the Rome Masters final for the second year in a row.
Australia's controversial tennis star Bernard Tomic has been labelled as the 'king of the excuse' by his country's media after he pulled out of the Rome Masters citing illness and excessive heat.
Novak Djokovic scraped over his first hurdle at the Rome Masters.
The 26-year-old then made first round exits from the Bordeaux Challenger and Geneva Open earlier this month.
Andy Murray is more afraid of losing his wedding ring than a tennis match these days but that did not stop him from storming to a 10th successive win on clay at the Rome Masters on Wednesday.
Sania Mirza and her Swiss partner Martina Hingis won their first ever title on red clay, capturing the Internazionali BNL d'Italia trophy.
Alcaraz has been seen as fellow Spaniard Rafael Nadal's successor on the Paris clay.
Jannik Sinner bounced back after losing the first set to beat Tallon Griekspoor in the third round of the Miami Open.
Maria Sharapova saved a match point to beat Li Na 4-6, 6-4, 7-6 (7/5) in the Italian Open final on Sunday after a two-hour rain break effectively turned the match into a tiebreak shootout.
Images from Day 2 of the 2024 French Open at Roland Garros in Paris on Monday.
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