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.
Italy, who have a maximum six points in Group J after a 2-0 win against Finland on Saturday, controlled the midfield through Verratti while Quagliarella set an example with his aggression.
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.
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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.
Christian Panucci was Italy's previous oldest goalscorer at 35 years 62 days.
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.
Italy next travel to Greece on June 8.
(Writing by Julien Pretot; Editing by Toby Davis)
