Jürgen Klopp has taken Liverpool to four European finals, one Champions League trophy and places that players and fans alike may never have imagined possible but a journey that enriched his reign is over. There will be no fairytale ending in Dublin after one more feat of escapology proved beyond Klopp’s team against Atalanta.

The 91st European game of Klopp’s nine-year Liverpool tenure was also his last despite a much-improved display and victory against Gian Piero Gasperini’s shaky, nervous but ultimately euphoric side. “If we fail, then let’s fail in the most beautiful way,” Klopp had said on the eve of the Europa League quarter-final second leg. Liverpool could not quite deliver. Mohamed Salah converted an early penalty that decided the game but could not transform the tie, however much Liverpool threatened to overturn a 3-0 first-leg deficit during a dominant first-half performance.

Related: Jürgen Klopp defends Salah’s finishing after Liverpool exit Europa League

Salah missed a great chance to double Liverpool’s advantage and heighten Atalanta anxieties. No better opportunities arrived in a second half in which Atalanta regained their composure to hold firm and reach a European semi-final for only the second time in their history. The Italian club celebrated one of their greatest nights long after the final whistle while Liverpool were left to rue the damage caused at Anfield last week. Their Europa League exit was largely self-inflicted and will gnaw away at Klopp. His team were firm favourites to win the one trophy they have competed for but not won during his reign, and the most expensive squad in the competition. One slim consolation was a performance that suggested all is not lost in the Premier League.

The improbable appeared probable when Liverpool were handed an early penalty by the Atalanta defender Matteo Ruggeri. Trent Alexander-Arnold also played an instrumental role in a lead that sent a wave of optimism through the visiting ranks and preyed on the worst fears of the hosts.

Alexander-Arnold was one of six changes Klopp made from the first leg, a tacit admission of getting things wrong at Anfield perhaps, and it was his exquisite pass out of central defence that sent Luis Díaz clear down the left. When the Colombia winger’s cross was headed clear by Ruggeri it was the Liverpool right-back who suddenly appeared to collect outside the penalty area. Alexander-Arnold’s cross struck the defender’s trailing hand and the French referee, François Letexier, was perfectly placed to spot the offence. Salah drilled the resulting spot-kick into Juan Musso’s bottom left hand corner as the Atalanta ­goalkeeper dived to the right. It was the seventh minute, the same time as Divock Origi had opened the scoring in the remarkable Champions League comeback against Barcelona.

Stadio di Bergamo witnessed a role reversal of the first leg at Anfield for 45 minutes. Now it was Liverpool playing with a vibrancy and clear sense of purpose as they swarmed over the Italian side. The lethargy of their last two defeats was gone. Atalanta were the ones making careless mistakes, posing little threat and being dragged into positions they did not want to go by Liverpool’s intelligent movement. The visitors threatened repeatedly and had eight attempts on goal, three on target, in the first half. Atalanta had three attempts on goal and none on target, although Teun Koopmeiners had a goal disallowed for offside shortly before the interval.

Roma beat Milan 2-1 to complete a 3-1 aggregate win and set up a semi-final showdown with the newly crowned Bundesliga champions Bayer Leverkusen. Gianluca Mancini gave Roma the lead in the 12th minute of the quarter-final second leg, the centre-back scoring on a rebound from close range.

Paulo Dybala doubled the hosts' advantage 10 minutes later with a curled shot inside the far post. However Roma lost striker Romelu Lukaku to an injury in the 28th minute and were left with 10 men from the 31st minute when the full-back Zeki Celik was sent off for a challenge on Rafael Leão. Matteo Gabbia pulled one back for Milan five minutes from time, nodding home a cross from Leão.

Marseille claimed a penalty shootout win over Benfica to reach the last four after Faris Moumbagna's late second-half goal levelled the tie at 2-2 on aggregate. Benfica's Ángel Di María missed the first kick of the shootout, striking the post, and the goalkeeper Pau López saved António Silva's effort to hand Marseille the win with all four of their penalty takers scoring.

The hosts broke the deadlock 11 minutes from time through Moumbagna's close-range header after Pierre-Emerick Aubameyang's cross from the left. The Ligue 1 side will take on Atalanta in their semi-final.

Salah almost released Dominik Szoboszlai through on goal only for Musso to gather at the midfielder’s feet. The keeper also saved at point-blank range from Díaz after another immaculate crossfield pass from Alexander-Arnold stretched the Italian defence again. The comeback looked possible.

Alexander-Arnold sent Cody Gakpo clear but overelaboration by the Liverpool forward line sent Klopp apoplectic on the sideline. His mood was not improved by Salah making a mess of an attempted lob over the stranded Musso after being sent clear by Gakpo’s first-time flick. Salah’s reaction, one of disgust, said everything about the miss. It denied Liverpool the bigger interval lead their performance deserved.

Atalanta created a rare opportunity to change the complexion of the second leg moments after the restart. Davide Zappacosta, the right wing back who was so effective at Anfield but largely contained here, presented Aleksei Miranchuk with a sight of goal inside the Liverpool area. The Russia striker’s shot was blocked but fell to Éderson, who steered a tame effort straight at Alisson.

Liverpool’s relentless efforts to turn the tie began to take a toll. Klopp introduced Darwin Núñez, Diogo Jota, Harvey Elliott and Joe Gomez in place of the tiring Díaz, Salah, Szoboszlai and Alexander-Arnold. The young striker Jayden Danns replaced Robertson as the Liverpool manager loaded his forward line. His team’s desperate pursuit of two goals also left defensive gaps that Atalanta started to exploit.

The changes failed to have the desired effect. Liverpool rarely threatened again as Atalanta rediscovered the defensive organisation that underpinned their victory at Anfield. Klopp’s European adventures petered out quietly.