If the search phrase isn't brand specific, I would assume this is due to relevant backlinks, dang good on-site optimization, and probably a fast loading, content heavy website for both the homepage and the service page. I have seen this before as well and if you were to do some in-depth competitor research on them, I believe you would find that it is more than likely a pretty well optimized website, and they have quality backlinks pointing to both the homepage and service page.
I would do a full analysis on their on-site factors (meta titles, outbound links, H2s, amount of content on both pages, internal linking, etc) and then use Moz Open Site Explorer or Ahrefs for both rankings pages to see what backlinks they have. Use that insight to see if you can, to a degree, duplicate what you see. Once you feel you have duplicated as much as you can, while still keeping the content original to your website, try and out-do them in both helpful, relevant on-page content and quality backlinks they don't have. Each situation is different and I am assuming some things here based on what I've seen in the past with this situation, hope it helps though and best of success!