Group E's surprise package went into this clash in second place, three points behind leaders Switzerland, but Norway's victory over Cyprus earlier in the night means they fall into third.
They were undone by a magnificent long-range strike from Kampl in the 19th minute and try as they might, they could not get themselves back into the contest.
It was the hosts who started in a positive manner at the Sportni Park, with Rene Krhin heading wide from a corner.
Their intent was rewarded when Kampl's piledriver from 25 yards rocketed into the top corner and past stranded Albania goalkeeper Etrit Berisha.
The Salzburg midfielder tried his luck from distance again a little later but was thwarted by Berish, while at the other end Slovenia goalkeeper Samir Handanovic made a comfortable save from Odise Roshi's speculative effort.
The Albanians went out for the second half with plenty of attacking intent, with Handanovic called upon to make fantastic saves from Lorik Cana's close-range effort and Hamdi Salihi's strike from just outside the area.
It proved to be Handanovic's day as he also kept out long-range shots from Edgar Cani and Valdet Rama to ensure Slovenia claimed their third win from seven group matches.
Their chances of emerging from this group would appear to be slim although leaders Switzerland's 4-4 draw with Iceland gives all the teams - bar minnows Cyprus - renewed hope of qualification.