This could be fire blight.  The most common point of infection is through the plant's flowers, however, leaves which are tattered or have tares from wind or hail can also be easily infected.   Pruning out affected branches once the bacteria are no longer active is the best way to manage the issue.   In addition to our factsheet, another good resource on how to execute this can be found here: http://ipm.ucanr.edu/PMG/PESTNOTES/pn7414.html

If only the leaves are affected i.e. you don't see stems darken and curve downward like a shepherd's crock then pruning may not even be necessary.   Cleveland select is noted for its resistance to the disease and it may not progress any further than the leaves. 

I hope this helps, please let me know if you have further questions.

Thank you Eric! We did some major pruning last weekend reaching as high as we could in an abundance of caution (and nervousness on my part) -- even on branches that just had leaves affected and not the branch. It may have been too much, but I think tree looks healthier and the browning does not seem to have spread. Thank you for responding!