It is just the second time Albania have qualified for a European Championship, after exiting at the group stage in 2016.
The hosts pushed hard for an equaliser after the break with Albania goalkeeper Etrit Berisha pulling off some fine saves, but in the 87th minute Vladyslav Baboglo’s low shot from close range finally gave the Moldova fans reason to celebrate.
Albania’s qualification effort began with a 1-0 defeat in Poland in March but they have since gone unbeaten, winning four games and drawing twice to reach 14 points, and can secure top spot with a win at home against lowly Faroe Islands on Monday.
Moldova are now in third place, level on 10 points with Poland and a point behind Czechia, and can still qualify.
They next face the Czechs away on Monday but if Czechia win Friday’s other group game with Poland in Warsaw then Moldova’s qualification dreams will be over.
Reuters contributed to this report.