Heroic Measures by Jill Ciment is a gentle 2009 novel about a very devoted long-married older couple who need to rescue their dog in New York City.  One of the charms of the books is that parts are told from the dog's point-of-view without personification.

One Good Dog by Susan Wilson is a 2010 novel about a man and dog who both need rescuing.  And although the reader just knows there HAS to be a happy ending, all good relationships and stories take time.