This work treats the problem of error-resilient DNA searching via oblivious evaluation of finite automata, where a client has a DNA sequence, and a service provider has a pattern that corresponds to a genetic test. Error-resilient searching is achieved by representing the pattern as a finite automaton and evaluating it on the DNA sequence, where privacy of both the pattern and the DNA sequence must be preserved. Interactive solutions to this problem already exist, but can be a burden on the participants. Thus, we propose techniques for secure outsourcing of finite automata evaluation to computational servers, which do not learn any information. Our techniques are applicable to any type of finite automata, but the optimizations are tailored to DNA searching.