"Finally" can be used to introduce the last item in a list, or the concluding argument in a discursive essay.
"Eventually" and "At last" can be used to introduce the final action in a story. They suggest that a goal or a point was reached after a lot of effort or hardship. ("Finally" can also be used in this scenario.)
Imagine an Oscar-winner �s speech:
I would like to thank my family and friends, my wonderful children, my drama teacher and, finally, my brilliant husband, Tom.
I never thought I would make it in this industry. I failed my first thirty-two auditions until eventually I was offered a small part in a production of �Macbeth � and at last I could do the job I had always dreamed of.