Another way of accomplishing this is to :w newName and then CTRL-^ (which is the same as CTRL-6) to switch to the new name.
When you :w to a new name, it sets that as the "alternate" file name, and CTRL-^ switches that to primary.  See :help alternate for more information on this.  Also useful is the :f newName which just renames the buffer (saving the old name as the alternate) without saving anything.
I like this a little better than the :saveas command because it doesn't introduce a new command, only a new shortcut, which is occasionally useful in other contexts as well.  It's always surprising to me that switching to the new name is not the default behavior of :w newName whenever I encounter it; it's hard for me to think of a use case where the existing behavior would be preferable.