The Argentina forward broke the La Liga goalscoring record by netting three times in a 5-1 thrashing of Sevilla on Saturday and he was at it again in Nicosia three days later.
Messi's struck after 38 minutes in the Group F encounter at the Neo GSP Stadium to pass Real Madrid legend Raul's record of 71 goals in Europe's premier club competition.
The mercurial Barca talisman's latest milestone came after Luis Suarez scored his first goal for the Catalan giants with a fine turn and finish as the Cypriot champions were totally outclassed.
Suarez had failed to find the back of the net in his first five matches since making his debut following a four-month ban for biting Italy Giorgio Chiellini during the World Cup, but his opening goal was worth the wait.
But it was Messi's night and he struck twice in the second half as Barca, who were already assured of a place in the last 16, eased to a victory that ensured their final group match against Paris Saint-Germain next month will decide who finishes top.
Barca had Rafinha sent off for a second bookable offence 20 minutes from time and defender Joao Guilherme was dismissed for the home side, who head to Ajax for their final game needing a win to secure a UEFA Europa League spot at the expense of the Eredivisie side.
Neymar and Xavi dropped to the bench as Luis Enrique made five changes to the side that hammered Sevilla, while APOEL striker Vinicius was passed fit after recovering from a knee injury.
Stand-in skipper Messi spurned a chance to surpass Raul after only two minutes as Urko Pardo saved his effort from the edge of the penalty area with his feet.
APOEL were inviting pressure as they sat far too deep and Jordi Alba should have made them pay when he surged forward and volleyed over the crossbar from close range after Messi picked him out with a sublime pass.
It was only a matter of time before Barca went in front and the opening goal came courtesy of a moment of class from Suarez after 27 minutes.
The Uruguay striker took a pass from Alba on the corner of the area and turned his marker with a lovely piece of skill before taking his time to beat Pardo with a measured right-footed finish to end his wait for a first Barca goal.
Messi then took centre stage yet again seven minutes before the break, when the recalled Rafinha tried his luck and the Argentine turned his shot beyond the wrong-footed Pardo to pass yet another landmark.
The visitors carried on where they left off after the interval and Messi took his Champions League goal tally to 73 with a clinical right-footed finish after Dani Alves had picked him out with an incisive pass 58 minutes in.
APOEL had offered nothing going forward, but were denied a first goal in four games when Marc-Andre ter Stegen produced an outstanding save to deny Tomas De Vincenti.
Rafinha was then shown a second yellow card for a challenge on De Vincenti, then APOEL were also reduced to 10 six minutes from time when Joao Guilherme was given his marching orders for a challenge on Pedro.
There was still time for Messi to add his third goal, applying the finish after Pedro had linked up with Xavi and cut the ball back for the man of the moment to score his second hat-trick in as many games.
