Messi became the first player to score five goals in a Champions League match as Barcelona thumped Leverkusen 7-1 at Camp Nou, sealing a 10-2 aggregate win in the last 16 and sending an ominous warning to its rivals with one of the competition's biggest ever victories.
"He is the best player in football's history and we've never seen anyone like him," Barcelona midfielder Cesc Fabregas said of Messi.
APOEL, meanwhile, continued the competition's most heartwarming story this season by beating Lyon 4-3 on penalties after winning the second leg 1-0 for a 1-1 aggregate score.
Goalkeeper Dionisios Chiotis saved two Lyon penalties in the shootout, sparking emotional scenes which saw APOEL players mount the goal frames in joyous celebrations with the team's fans.
While APOEL's was a big-hearted team display to overcome one of the Champions League's more established sides, Barcelona's huge victory was down to one man.
Messi was unstoppable, scoring twice in the first half and three times after the break to move to 49 competition goals — joint fourth on the all-time European Cup list alongside Alfredo di Stefano.
With 12 Champions League goals this season, the Argentina forward equaled the record for one campaign he holds jointly with former Manchester United and Real Madrid striker Ruud van Nistelrooy.
"He's the best (ever), there is no other like him," Barcelona coach Pep Guardiola said.
Granted the freedom of the Nou Camp by the beleaguered German visitors, the three-time world player of the year took full advantage to leave Guardiola and the Barca fans shaking their heads in disbelief.
His deft chip over Leverkusen goalkeeper Bernd Leno put Barca ahead in the 25th minute before Messi collected a pass from Andres Iniesta, cut inside to fool a defender with a dummy and curl a finish into the bottom corner.
Messi completed his hat trick with another chip with his weaker right foot in the 49th and added further goals in the 58th and 85th minutes, leaving Leverkusen's players on their knees.
"His class is just extraordinary, there's no discussion," Leverkusen coach Robin Dutt said.
Substitute Cristian Tello's brace will get lost in the rush to hail Messi, who has scored 53 goals in 49 games for club and country this season.
With its small €10 million ($13.2million) budget, APOEL had already exceeded expectations by reaching the knockout stage, but the fairytale journey will go on for at least two more matches.
Gustavo Manduca, who was later sent off for two bookable offences, levelled the series at 1-1 on aggregate in the ninth minute and APOEL almost didn't need penalties when Lyon defender Aly Cissokho's attempted clearance hit his own bar.
Chiotis saved from Alexandre Lacazette and Michel Bastos in the shootout while APOEL converted all its kicks, setting off wild celebrations inside the ground of the tiny east Mediterranean island.
Barcelona and APOEL join AC Milan and Benfica in the quarterfinals, with the rest of the last-16 matches completed next week.