If you can finish your masters and get a degree from a UK university, it will be MUCH easier for you to qualify for a work visa than if you just quit the masters (in fact, it's likely to be next to impossible to qualify for a work visa without completing your masters).
Without a UK degree, you won't be able to get sponsorship unless:
a) the job is listed on the skills shortage list (usually these are higher-up, management-level jobs for people with several years of experience)
or
b) the UK company has advertised the job across the UK and entire EU for a certain amount of time and have found no suitable candidates for the job (the Resident Labour Market Test). With unemployment still high and 500 million people in the EU, this can be almost impossible.
But if you have a degree from a UK university and you are still in the UK on a valid Tier 4 visa once you have finished the degree, you don't have to meet either of these requirements. The company can sponsor you WITHOUT the job needing to be on the shortage list and WITHOUT needing to pass the Resident Labour Market test.
Also, if you haven't finished your degree yet, I believe you will have to return to the US to apply for a work visa - as I understand it you can't switch from a Tier 4 visa to a Tier 2 work visa in-country if you don't have a UK degree.
See:
https://www.gov.uk/tier-2-general/switch-to-this-visaand
http://www.ukcisa.org.uk/International-Students/The-next-stage/Working-after-your-studies/Employment-Tier-2/Resident-Labour-Market-Test-and-exemptions/Alternatively, you mentioned in a previous post that you have an Irish boyfriend - if you were to marry him and apply for an EEA 5-year residence card, you wouldn't need to be sponsored for a work visa at all - you would have the right to live and work in the UK, in any job you like, without needing a work visa.