Selon le TLFi :

Du lat. pop. *rasare, altér. (d’apr. le supin rasum) du class. radere « raser », « gratter, polir, ratisser » et « toucher en passant, effleurer, côtoyer ».

All Slavic languages share the verb rězati which means to cut.

Their etymon is assumed to be *razЪ which would come from *rōzo but there is no agreement about its Indo-European origin, if any. Some authors suggest it is *u̯reĝ-.

On the other hand, raser comes for the Latin radere which relates to the Indo-European *reh₁d-.

There is then no evidence of any link between these verbs.

References:

• HAL Actualisation, intensité et saillance : raz mot autonome
  et préfixe en russe contemporain
  Olga Kravchenko-Biberson

• Wikipedia raser, rado, rodo