How should I describe a result from a submitted paper such that the reviewers can do their work?
When submitting a paper, you are normally able to provide attachments for consideration by the reviewers. You can include a draft of the submitted manuscript as a reference for the readers.

However, if you're in mathematics or another field which encourages preprints, you should in principle be able to post the paper on a repository like arXiv and point the reviewers to the repository version while the paper is under consideration. If the paper is accepted and appears in print before the new paper is accepted, you can change the reference in the proof stage.

What you can get away with depends upon the editorial policies of the journal. Some journals require you to provide copies of any material that is unpublished (such as correspondences with other researchers or unpublished manuscripts). Other journals will let you get away with less. But in general, you need to show some sort of evidence that the work is under consideration. Perhaps an email confirmaing submission, plus the abstract, will suffice?