Cole was at the heart of his new side's best moves and set up the opening goal for David Ngog.
Captain Steven Gerrard slotted home a penalty just before the interval and after that it was more of an exhibition game as Liverpool eased into the play-off draw.
The Macedonians put up little in the way of opposition and the visit of Arsenal in little over a week's time for the opening fixture of the domestic season will be a completely different proposition for the Reds and their new boss Roy Hodgson.
However, he can take heart from the performance of Cole on his debut.
The 28-year-old was handed the central attacking role he so craves and did plenty to justify his claims for taking that position off Gerrard, who has been so effective playing behind the striker in recent seasons.
Hodgson adopted the 4-2-3-1 formation used by his predecessor Rafa Benitez rather than the 4-4-2 he used at Fulham but the system looked more fluid than it had last season.
Ngog had the ball in the net early only to be flagged offside but he fluffed his first genuine chance in the 14th minute.
A backheel by Cole released Gerrard who put last week's two-goal hero clean through and although Martin Bogatinov dropped to his knees far too early the French striker took the ball too close to the goalkeeper who cleared.
The precision of Cole's lofted pass was not matched by the first touch of Spain Under-19 international Dani Pacheco, making his first start, but the opening goal was not long in coming.
A low 25-yard shot by Gerrard was tipped around the post by Bogatinov and Liverpool took the lead from the resulting 21st-minute corner.
The captain played the ball short to Cole who curled a cross towards the penalty spot where Ngog headed in off the underside of the crossbar.
Five minutes before half-time Fernando Lopes fouled Ngog as he turned Lucas Leiva's looping header past the defender and Gerrard slotted home from the penalty spot.
Less than 40 seconds after the restart Cole's cross-shot rebounded off the angle of crossbar and post while another new signing Milan Jovanovic's low shot was palmed around his left-hand upright by Bogatinov.
Cole then forced the goalkeeper into a low save to his left, Pacheco fired in a shot from the left wide of the far post and Ngog headed tamely at Bogatinov as the one-way traffic continued.
Gerrard, who last week committed his future to the club despite speculation to the contrary, soon departed to a standing ovation to be replaced by Alberto Aquilani while Jovanovic, who had also flashes of some much-needed creativity, made way for Maxi Rodriguez.
But the focal point continued to be Cole, whose display deserved a goal, as he was denied at the near post again.
Yet another Cole pass played Rodriguez in on the right of the penalty area and he cut the ball back on to his left foot but his shot was blocked by Egzon Belica.
Cole was destined not to get on the scoresheet as Martin Kelly's left-wing cross was too hard and high for the unmarked diminutive midfielder to get a proper header on.
However, Liverpool, who go into tomorrow's draw in Nyon for the play-off round, will be more than happy if Cole saves his strikes for more auspicious occasions.
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