This seems utterly bizarre: if I have rights to edit posts unilaterally without approval, shouldn't I still have rights to approve low-rep users' edits unilaterally as well?
Yes.
And you do. Just click "Improve" when reviewing the edit, make some minor change (or lots of major changes, if the edit you're approving didn't go far enough), and click Save Edits. The pending edit will be approved instantly, and your edit submitted immediately after.
Of course, your name is on the post now, but that shouldn't matter if the edit was a good one.