Regular expression can be represented by a finite automata and more precisely by a Deterministic one.
When you write a Regex, the lexer will convert it into a DFA to find matches in the text. So of course finite automata have its role in the lexical analyser.
There are very simple algorithms to convert Regex to FA like the Thompson's construction algorithm (http://en.wikipedia.org/wiki/Thompson's_construction_algorithm) but you can find optimized algorithm.