Sort of, but not in the same way that modern white supremacists think of 'superior race.'

Romans, like Greeks, divided the world into civilized peoples and barbarians. Civilization was related to one's birth into a kin-group (the Roman words for 'people,' natio and gens, both refer etymologically to someone's birth: think 'natal' and 'generate'), but it was also about education and behavior. A civilized person spoke a certain way, used certain kinds of gestures, and knew certain kinds of knowledge. These values were rooted in Greek traditions, reinterpreted within the Roman empire. Barbarians, on the other hand, talked funny, dressed funny, ate funny foods, and had weird local customs. The lines between civilization and barbarians were often blurred, and barbarians could enter the empire and, if they behaved properly, pass as Romans pretty effectively and even become a Roman citizen (though, like first generation immigrants in the US in the 20th century, this passing could always be challenged, and we have many examples of barbarians who weren't able to shake their outsider status). On the flip side, Romans, especially members of the military, were often accused of having become barbarians by members of the educated, wealthy, civilian class, because military dress and manners blended extensively with the dress and customs of people living on the frontiers, and military men didn't fit in well with civilian culture.

The difference between Roman (or Greek) and barbarian was very imagined in very hierarchical terms. Romans were 'real' men, and barbarians were somewhere between humans and animals. Barbarians could be noble savages when it served a Roman author's rhetorical needs, and they could just as easily be wild beasts; but they were never considered equals (the exception often being Persia, which Romans had to grudgingly admit to be something of a political equal, and hence something more than the typical barbarians).

So in these terms, the Romans definitely considered themselves to be the superior people. But unlike modern racism (which defines race in terms of skin color, skull shape, and outdated theories of human evolution), Roman superiority was rooted in membership in an imagined kin-group, education, manners, language, and citizenship. You didn't have to look Greek or Italian to pass as civilized; but if you talked with a funny accent, and had parents from Germania, people might start to question whether you really belonged.

(And note the implication of that last point: with civilization being tied to accent and education, a lot of poor Romans - who were citizens after the mid 3rd century - were technically insiders in the empire and not barbarians, but still couldn't rise very far because they lacked the skills of a civilized person; racial/ethnic discrimination and social class always go hand in hand.)