We use GitHub flow as branching strategy which means that anything in master is the result of a merged pull request. It's trivial to get the last commit by doing the following.
GET https://api.github.com/repos/:org/:repo/git/refs/heads/master
However it seems non-trivial to work out the pull request that caused this commit.
Is it possible to find out the pull request that created this commit via the API?
The PR should be one of the parents—the second one—of the commit in master, since that commit is the result of a merge of the PR.
You can find that parent with the GitHub commit API
GitHub doesn't offer a way to get the branch from a commit, but you could list all the PR and cross-reference the SHA1 found above with said PRs.