Pique nodded home in the 28th minute to help Barca - and their new head coach - kick off their Group F campaign with maximum points, although the determined Cypriot side offered a sterner test than many had predicted.
The Spanish giants' triumph ultimately continued a 100 per cent start to the tenure of Enrique, who took the reins following Gerardo Martino's ill-fated campaign at the helm in 2013-14.
Having managed to once again align a defensive nous with the kind of possessional dominance that was the hallmark of the Pep Guardiola years, Barcelona are beginning to look imposing.
Unsurprisingly, they had too much for APOEL, with Barca's excellent defence - who have yet to concede this term - yielding only a single shot on target as Enrique's men followed up La Liga wins over Elche, Villarreal and Athletic Bilbao.
Any thoughts that the new coach might take APOEL lightly were soon put to rest as Lionel Messi, Neymar and Munir El Haddadi started together up front.
Messi was on hand immediately to illustrate his threat, forcing goalkeeper Urko Pardo into a solid save with a dipping free-kick.
That set the early tone as Barcelona controlled possession, with Xavi - who has become more of a bit-part player under Enrique - at the centre of their good work.
Their impressive build-up play did not glean too many chances, although Neymar did curl a shot just wide midway through the opening period.
However, the deadlock was finally broken in the 28th minute, Pique finding space in the area to nod Messi's pin-point free-kick into the corner.
There was little sign of an APOEL response, with Barca quickly settling back into a methodical passing rhythm.
And the Spanish giants should have gone two up in the 40th minute when Messi uncharacteristically hit straight at Pardo following Neymar's pull back.
Picking up where they had left off, Barcelona pressed forward further at the start of the second period.
But the visitors continued to defend impressively in numbers, with Enrique's side struggling to find a way through their rearguard.
APOEL managed to offer the odd moment of alarm for the hosts, too, even if their final ball left plenty to be desired.
The Spaniards brought Andres Iniesta on for his long-time partner in midfield Xavi, yet even his introduction could not bring a second.
Barca were again denied just before the 90-minute mark, substitute Sandro shooting narrowly wide from just outside the area.
That seemed set to be the catalyst for a thrilling finale as Messi saw an effort cleared off the line by Mario Sergio before, at the other end, Marc-Andre ter Stegen was forced into his first save in the dying moments.
The goalkeeper reacted smartly to turn away Gustavo Manduca's effort and snuff out hopes of a famous draw for the visitors.
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