Quagliarella, 36, netted two first-half penalties, while 19-year-old Moise Kean scored his second goal in his third appearance. Stefano Sensi, Marco Verratti and Leonardo Pavoletti also found the net.
Liechtenstein were reduced to 10 men on the stroke of halftime when Daniel Kaufmann was sent off for a handball on his goalline.
The Azzurri, who are rebuilding under Mancini after missing out on the 2018 World Cup, broke the deadlock after 17 minutes as the 23-year-old Sensi's header from Leonardo Spinazzola's cross from the left found the roof of the net.
Fabio Quagliarella of Italy celebrates his goal of 3-0 with teammates Frello Filho Jorge Luiz Jorginho, Gianluca Mancini and Leonardo Bonucci during the 2020 UEFA European Championships group J qualifying match between Italy and Liechtenstein at Ennio Tar
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Verratti doubled the tally with a fine curled shot into the far side of the goal on 32 minutes.
Quagliarella scored his first goal from the penalty spot after a Nicolas Hasler handball, and his second after Kaufmann also used his arm and was shown a straight red card.
Kean, who had endured a frustrating first half, headed home the fifth on 70 minutes with second-half substitute Pavoletti adding the sixth on his international debut, tapping in after Benjamin Buechel had parried the 30-year-old's header into his path.
Denmark’s Henrik Dalsgaard scored a 93rd-minute equaliser to complete an incredible comeback from three goals down to draw 3-3 away against Switzerland on Tuesday in a pulsating Euro 2020 qualifier in Basel.
Denmark's players celebrate their team's third goal during the UEFA Euro 2020 Group D qualification football match between Switzerland and Denmark at the St Jakob Park stadium in Basel on March 26, 2019.
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Switzerland were on the verge of making it two wins out of two at the start of their Euro 2020 qualifying campaign but Dalsgaard outjumped goalkeeper Yann Sommer to head home and earn Denmark an astonishing and unlikely point.
The home team broke the deadlock in the 19th minute when Remo Freuler beat Denmark keeper Kasper Schmeichel with a cool finish for his first international goal.
Swiss captain Granit Xhaka hammered home a 20-metre shot in the 66th minute and Breel Embolo made it 3-0 from close range after Manuel Akanji jumped highest to head across goal.
Yet Denmark came back from the dead in the dying moments of the match with goals by Mathias Jorgensen, substitute Christian Gytkjaer and Dalsgaard in less than 10 minutes.
(From L) Denmark's defender Jens Stryger Larsen, defender Henrik Dalsgaard and forward Pierre-Emile Hojbjerg celebrate their team's third goal during the UEFA Euro 2020 Group D qualification football match between Switzerland and Denmark at the St Jakob P
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Norway substitute Ola Kamara scored in the 96th minute to snatch a 3-3 draw at home to neighbours Sweden in a fiercely-contested Group F Euro 2020 qualifier on Tuesday.
Robin Quaison had scored two late goals to give Sweden a 3-2 lead in stoppage time but Kamara headed home with virtually the last touch of the game to snatch a point in a pulsating game.
The result leaves Sweden in second place on four points from two games, two points behind leaders Spain who won 2-0 away to Malta, while Norway are fifth on one point.
The Nordic rivals traded thunderous tackles in front of a packed Ullevaal Stadium, with Josh King bringing the home crowd to their feet early on as he hit the bar with a towering header.
Norway´s Martin Odegaard reacts after missing a chance during the UEFA Euro 2020 qualifying football between Norway and Sweden in Oslo on March 26, 2019
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King was involved again as Norway took the lead when his shot was blocked into the path of Bjoern Johnsen, who fired home in the 41st minute from what appeared to be an offside position.
King scored himself in the second half, winning the ball from Albin Ekdal in the centre of the pitch to start a counter-attack before racing into the box to head home Markus Henriksen's cross in the 59th minute.
The Swedes got a lifeline in the 70th when Viktor Claesson was bundled over in the area and then rattled home the rebound after Andreas Granqvist's penalty was saved by Rune Jarstein.
They levelled in the 86th when Quaison's shot took a wicked deflection off defender Haavard Nordtveit and, as the clock ticked past 90 minutes, Quaison then steered Classon's pass beyond Jarstein to put the Swedes aheead.
Sweden´s players celebrate scoring during the UEFA Euro 2020 qualifying football between Norway and Sweden in Oslo on March 26, 2019.
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The drama did not end there and Jarstein made his way forward as Martin Odegaard took a corner deep into stoppage time, with Kamara popping up to nod home a late equaliser for Norway in a thrilling climax to the game.
Greece fought back from two goals down to earn a 2-2 draw in Bosnia after the hosts had Miralem Pjanic sent off in a pulsating Euro 2020 Group J qualifier on Tuesday.
The result left both sides on four points from two games, two behind leaders Italy who thrashed Lichtenstein 6-0 at home and one ahead of Finland after their 2-0 win in Armenia.
The Armenians and Lichtenstein have no points.
Bosnia took a 2-0 lead through Edin Visca and Pjanic in the opening 15 minutes but Konstantinos Fortounis pulled one back with a penalty and substitute Dimitris Kolovos levelled after Pjanic was shown a straight red card for a reckless tackle on Zeca.
Miralem Pjanic (2nd L) of Bosnia and Herzegovina celebrates with his teammate after scoring during UEFA Euro 2020 Group J qualifying match between Bosnia and Herzegovina and Greece at the Bilino Polje Stadium in Zenica, Bosnia and Herzegovina on March 26,
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Roared on by a fervent home crowd, the Bosnians made a flying start as Visca steered in a left-footed shot before Pjanic left Greece keeper Odisseas Vlachodimos rooted to his line with a superb free kick from 30 metres.
Fortounis drilled in his 64th-minute spot-kick after he was tripped by Toni Sunjic and Pjanic turned from hero to villain barely 60 seconds later after a studs-up challenge on Portuguese-born midfielder Zeca.
Bosnia keeper Ibrahim Sehic denied Kolovos with a reflex save in the 77th but was powerless in the 85th as the midfielder drove home an unstoppable header off the underside of the bar after a fine cross from the right by Zeca.
In the day's early fixture, close-range goals by Fredrik Jensen and Pyry Soiri gave Finland victory in Armenia, with both sides missing chances in an action-packed encounter.
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