This question is a lot more complex than it seems. In short, the answer is that we cannot know with certainty.

There are two broad schools of thought. The first, and older, narrative is that the Lombards migrated into Italy and imposed themselves over a largely subservient Latin population which itself had been depleted and whose institutions were destroyed thanks to both plague and decade-long war between the post-Roman Kingdom of Italy and the Eastern Empire. The second, and more modern, interpretation aligns the Lombards with the earlier tradition of recruitment of non-Latinized peoples into armed forces, as was commonplace in the Late Empire and Italian states would largely continue to do (to some degree) until the unity of Italy.

As you might imagine, sources on the superposition of the Lombard ruling class onto the existing Romano-Italian society are conflicted; while some Latin sources are not kind to the Lombards, there is also an incredibly rapid integration of the Lombard warrior-aristocracy and the old Roman "Senatorial" class. Hardly the evidence of a cultural and political domination. In fact, the Lombards were latinized much faster than the embryonic Italian language (or rather, local vernacular) absorbed Lombard-isms. Sure, the Lombards had their own separate legal system, but the only sources that exist on them are in Latin, and they deal almost exclusively with the resolution of high level-confrontations, blood feuds, and "Private Wars." In short, Lombard laws almost seem like a "Warrior's Code" of sorts, while everyone else continued to use the Latin legal system, enforced up by the urban institutions traceable to the days of the Roman Empire. By the following century, the word "Lombards" had come to mean "Italian not living under the Byzantine Empire."

The latest literature (up to and including including researched published by Massimo Montanari at the University of Bologna in 2015) can only agree that there no real consensus as to how or why the Lombards entered Italy so easily and integrated with the old roman senatorial aristocracy. The narrative of domination is also done no favors by the estimated numbers of "Lombards" that entered Italy (with the high end of estimates never surpassing 10% of the Italian population, while the real number was probably much lower). However, it's also true that the Emperor in Constantinople who nominally governed Italy certainly had other issues to deal and couldn't or wouldn't defend the unruly peninsula he had reconquered at enormous cost, while internal instability and manpower problems within the Italian governing class itself certainly facilitated the embrace of a foreign "Warrior Society" which rapidly integrated with the Latin ruling class (much like the Normans in southern Italy would do five centuries later).