Albania took the lead in the 38th minute when Lenjani surged through the middle and unleashed a speculative shot that clipped Simon Kjaer's leg and wrong-footed Kasper Schmeichel in the Danish goal before nestling in the bottom corner.
Despite Denmark's pedestrian passing and toothless attack, they almost equalised 10 minutes into the second half when substitute Vibe's low cross was turned onto the post by Albania captain Lorik Cana.
Lenjani then wasted a glorious opportunity to double his tally for the home side, dragging a left-footed shot agonisingly wide of the far post after turning Kjaer inside out.
The Danes equalised in the 81st minute when Nicklas Bendtner slalomed into the area and was upended for what looked like a certain penalty, but Vibe was on hand to slam the loose ball home before the referee had a chance to blow his whistle.
The result means Denmark top the five-team group on four points, the same as Albania but with more goals scored. Serbia, who host Albania next week, and Armenia both have one point after drawing earlier on Saturday.
Portugal, who visit Copenhagen to play Denmark on Tuesday, are bottom of the group having lost their only game to date against Albania.
(Reporting By Philip O'Connor)
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