Italy’s only remaining representative in Europe came into the clash at the Stadio Olimpico with a slender lead but two away goals from last week’s first leg in Amsterdam.
Four-time European champions Ajax dominated from the off but in the opening period struggled to break Roma down despite monopolising possession.
However that changed once teenager Brian Brobbey was introduced for Antony at the break.
The 19-year-old striker, who made his Eredivisie debut earlier this season, made an instant impact when he raced onto Perr Schuurs pass three minutes after the restart and nipped the ball past an onrushing Pau Lopez.
Dusan Tadic thought he had put Ajax ahead in the tie six minutes later only for his neat finish on the rebound from a Lopez save to be ruled out for a foul on Henrikh Mkhitaryan in the build up.
And with Ajax looking like steamrolling their opponents Edin Dzeko tipped the tie in Roma’s favour with 18 minutes remaining.
The Bosnian was on hand to tap home the leveller which put Roma back in the lead on aggregate after Riccardo Calafiori’s low cross was deflected into his path.
With Ajax needing to score again to take the tie into extra-time the hosts were galvanised and managed to fend off Erik ten Hag’s outfit, who will have to wait until next season for a shot at European glory.
Roma will be hoping to gain revenge on United for the 7-1 hammering at Old Trafford at the quarter-final stage of the Champions League in 2007.
United, Arsenal stroll through
United had few problems against Granada, who had a mountain to climb after losing the first leg 2-0 in Spain.
Edinson Cavani opened the scoring after just five minutes to put United ahead on the night when he lashed home first time Paul Pogba’s looping knockdown to follow up on his strike against Tottenham in the Premier League at the weekend.
The away side pressed and had some chances but Jesus Valleho’s last-minute own goal sealed the win for United, who could yet have an all-English final with Arsenal.
The Gunners will face Villarreal in a fortnight after cruising past Slavia Prague 4-0 to go through 5-1 on aggregate.
Behind on away goals thanks to Tomas Holes’ stoppage-time leveller at the Emirates Stadium last week, Arsenal ended Slavia’s 23-match unbeaten run thanks to a six-minute blitz in the first half that sealed the tie.
Emile Smith Rowe had already had a goal ruled offside by VAR when Nicolas Pepe put Arsenal ahead on the night and in the tie in the 18th minute.
The 20-year-old Smith Rowe pushed Ivorian Pepe through to coolly finish and calm any Arsenal nerves.
Alexandre Lacazette made no mistake from the spot three minutes later after Bukayo Saka was brought down and with 23 minutes on the clock Saka made it three with a driving run and drilled near post finish.
Lacazette scored his second of the night in the 77th minute, collecting Pepe’s low cross before drilling home a left-foot finish that put the cherry on the cake for Mikel Arteta’s side.
Unai Emery is on course to claim his fourth Europa League crown as a coach after his Villarreal team made short work of Dinamo Zagreb in a 2-1 win to qualify 3-1 on aggregate.
Paco Alacer opened the scoring nine minutes before the break when he finished off a superb team move and Gerard Moreno continued a career-best season shortly afterwards, firing home his sixth goal in the competition after Dominik Livakovic’s save.
Mislav Orsic salvaged some pride for Dinamo with 16 minutes left, but it was to little too late to save the Croatian champions’ European campaign.