to file for H1  you need atleast a  Bachelors degree. I am not sure if you are currently working on CPT/internship or OPT(Associates) and also not sure if the job you are doing is related to the field of your study. But  the trick here is that the job requirement can be changed....

That being said even if you applied for the job which did not originally require Bachelors degree; and if your company supports, they can change the requirement and also your position (which has to be related to your field of study).   you should talk to the HR and attorney who will be taking your case to adjust all the requirements necessary to file for H1. 

For example if you are a computer science major and if you submit your resume which has only lab processing of samples, doing experiments, aliquoting serum/DNA etc and you don't mention doing anything with or in computer chances are it wont be approved, so you should adjust your resume as keeping records of subject/sample info in the lab computers/database...... exporting records through excel to provide data to research scientists....... generating weekly reports etc etc ........ then it will support your Computer Science major but since I don't know what your major is, so you need to take care of these minor things and i am pretty sure your H1 will be approved.