Yes
Awarding experience as entire levels instead of keeping track of the experience points is an easy and reasonably popular alternative way for character advancement. The Dungeon Master's Guide has a small section devoted to this variant, "Level Advancement Without XP", on page 261.
You can award levels on a per-session basis, or based on campaign progression. In my own DnD 5e games, we use the latter: whenever the players complete an adventure or a particularly dangerous part of one, they gain a level at the end of the session.
An advantage of this method is that it is possible to fine-tune and pre-plan the experience progression very easily to suit your tastes, whether that means a swift "zero-to-hero" ascension or an extended adventuring on the lower levels. There's also less need for bookkeeping, and the players may feel less pressure to seek out combat and other challenges just for experience. A possible disadvantage is that the progression can feel arbitrary and less objective than XP, and removes the immediate experience reward for succeeding in combat or other challenges.