No, the level of your Pokémon does not directly increase the chance to successfully run away (as of gen 5+).
Source 1:
On this Bulbapedia article, it says that this is the equation to determine your chances to successfully escape:

Neither this nor the "other factors" section under it mentions the level of either Pokémon. See the linked "other factors" or the linked Pokébase posts in sources 2 and 3 for more information on what does affect the chances of running away successfully.
Souce 2/3:
Neither This post nor this post mentions the level effecting your chances of escaping successfully. The BA for the second one specifically mentions that it doesn't.
Source 4
Experience
Hope I helped!