Mane scored a second for the Champions League winners in the early stages of extra time, but Chelsea hit back with a Jorginho penalty to force spot kicks, which Jurgen Klopp's side won 5-4 to triumph.
It proved to be a special night for referee Stephanie Frappart, who became the first female in history to officiate in a major men's European match.
Liverpool had the game's first half-chance in the fifth minute when Mane met a high ball inside the Chelsea box, but the attacker's bicycle kick was blocked by Andreas Christensen. Joel Matip then headed into the arms of Kepa Arrizabalaga moments later as the Reds made a decent start to the contest.
The end-to-end nature of the match continued as the play developed, Kepa making a smart save to keep out a low Mohamed Salah effort following smart work from Alex Oxlade-Chamberlain before Giroud sent one wide of Adrian's post moments later.
Chelsea had a golden chance to take the lead in the 22nd minute when Pedro broke into the Liverpool box before striking towards goal, only for the ball to smash off the crossbar as the woodwork frustrated the Blues for the second match in a row.
Liverpool were next to come close just before the half-hour mark when Virgil van Dijk climbed highest inside the Chelsea box to meet a corner from Salah - Kepa got nowhere near it but the centre-back's effort landed over the crossbar, before Mane headed into the arms of the Chelsea goalkeeper moments later.
Lampard's side remained a huge threat down the other end, though, and Adrian had to deny Mateo Kovacic before Joe Gomez made an excellent challenge to prevent Giroud from potentially sliding a low Pedro cross into the back of the net.
The London club were not to be denied, though, and they made the breakthrough in the 36th minute when Giroud collected a fine pass from Pulisic before picking out the bottom corner with an excellent finish.
Pulisic thought that he had scored the second goal of the Lampard era five minutes later after picking out the bottom corner with a brilliant finish following a mazy run, only for the offside flag to come to Liverpool's rescue.
Liverpool made a change at the break with Roberto Firmino replacing Oxlade-Chamberlain, and it did not take long for the Brazilian to have an impact; it was his assist that allowed Mane to level the scores in the 48th minute with the Senegal international converting from close range after Kepa had missed the ball.
Fabinho then whistled one wide of the Chelsea post moments later after more good work from Firmino, who made a huge difference to the Reds in the final third of the field.
Klopp's side had another big chance in the 52nd minute when Mane broke into space before feeding Jordan Henderson - the midfielder's effort was deflected towards goal but Kepa managed to make a smart save.
Georginio Wijnaldum was the second player off the Liverpool bench just past the hour, and the alteration came during a strong spell for Chelsea, who had managed to ride out the spell of pressure at the start of the second half without conceding a second goal.
Liverpool had a golden chance to score in the period that followed as the ball dropped for Van Dijk inside the Chelsea box following a save from Kepa, but the Spanish goalkeeper somehow managed to push the Dutchman's close-range attempt onto the post as it stayed level in Istanbul.
Salah then saw a free kick deflected over the crossbar before Van Dijk blocked an attempt from Pedro down the other end with both teams looking for the game's third goal.
Mount did have the ball in the back of the Liverpool net in the 83rd minute courtesy of a smart finish, but again the offside flag came to the Merseyside club's rescue.
As it turned out, the two teams could not be separated at the end of normal time despite a late Salah half-chance, meaning that extra time and potentially penalties would be needed to decide the winner of the 2019 UEFA Super Cup.
Neither Klopp nor Lampard would have been enthused at the thought of an additional 30 minutes at this stage of the season. Liverpool had the first half-chance of the additional period when Gomez broke into a shooting position, although it was a comfortable save for Kepa.
The Reds took a 2-1 lead in the 95th minute when Mane thumped a low pass from Firmino into the back of the net off the crossbar as the clock struck midnight in Istanbul.
Chelsea were level just six minutes later, however, with Jorginho converting from the penalty spot after Adrian had brought Abraham to the ground inside the box.
Abraham had a golden chance to make it 3-2 Chelsea in the latter stages of the first half of extra time, but the striker turned a low Pedro cross wide of the post from close range. The Blues also had the best two chances of the second half through Mount and Pedro, but penalties were ultimately needed.
A high-quality shootout saw the first nine players find the back of the net, but Abraham's missed attempt at 5-4 saw Liverpool land more European silverware.
Klopp's side will attempt to make it back-to-back Premier League wins at the start of the new season when they travel to Southampton on Saturday, while Chelsea will be looking for their first victory of the campaign at home to Leicester City on Sunday.
LIVERPOOL: Adrian; Gomez, Matip, Van Dijk, Robertson (Alexander-Arnold 91'); Fabinho, Henderson, Milner (Wijnaldum 64'); Oxlade-Chamberlain (Firmino 46'), Mane (Origi 103'), Salah
CHELSEA: Kepa; Azpilicueta, Christensen (Tomori 85'), Zouma, Emerson; Kovacic (Barkley 101'), Jorginho, Kante; Pedro, Giroud (Abraham 74'), Pulisic (Mount 74')