You need to pass the --keep-cr flag to git am.  It's unfortunate, but because of competing standards (email workflow versus local), there's really not much choice.
You may want to try setting up a .gitattributes file as well.  Trying to recreate your issue, I was able to get things to work when I specified the file as requiring CRLF.  Note, without normalizing the files first, it did show the whole file as modified.  I commonly use a .gitattributes with this:
.gitattributes  export-ignore
.gitignore      export-ignore
*.txt           text
*.C             text trailing-space space-before-tab -indent-with-non-tab
*.rst           text trailing-space space-before-tab -indent-with-non-tab
*.clj           text trailing-space space-before-tab -indent-with-non-tab
*.c             text diff=cpp trailing-space space-before-tab -indent-with-non-tab
*.cpp           text diff=cpp trailing-space space-before-tab -indent-with-non-tab
*.h             text diff=cpp trailing-space space-before-tab -indent-with-non-tab
*.hpp           text diff=cpp trailing-space space-before-tab -indent-with-non-tab
*.py            text diff=python trailing-space space-before-tab -indent-with-non-tab
*.tex           text diff=tex
*.java          text diff=java trailing-space space-before-tab -indent-with-non-tab
*.pl            text diff=perl trailing-space space-before-tab -indent-with-non-tab
*.php           text diff=php
*.rb            text diff=ruby trailing-space space-before-tab -indent-with-non-tab
*.vcproj        eol=crlf
*.dsp           eol=crlf
*.dsw           eol=crlf
*.sh            eol=lf
*.jpg           binary
*.png           binary
*.gif           binary
*.tiff          binary
You'll want to normalize your line ending according to the gitattributes man page.  Another SO user ended up turning off core.autocrlf as well to get clean commits and patches.
Trying to reproduce your bug
$ git init repo
Initialized empty Git repository in c:/tmp/git-eol/repo/.git/
$ cd repo
$ git config --local core.autocrlf false
$ vim foo.txt
$ git add foo.txt
$ git commit -m "Add foo."
[master (root-commit) 3903abd] Add foo.
 1 file changed, 3 insertions(+)
 create mode 100644 foo.txt
$ vim foo.txt
$ git st
## master
 M foo.txt
$ git commit -m "Add more foo." -a
[master 03e991a] Add more foo.
 1 file changed, 2 insertions(+)
$ git format-patch HEAD~1
0001-Add-more-foo.patch
$ vim 0001-Add-more-foo.patch
Looking at the patch file that was produced, I see this:

As you can see, the carriage returns (^Ms) were kept in the patch.  This is with core.autocrlf=false.  Continuing on, I see:
$ git reset --hard HEAD~1
HEAD is now at 3903abd Add foo.
$ git am 0001-Add-more-foo.patch
Applying: Add more foo.
error: patch failed: foo.txt:1
error: foo.txt: patch does not apply
Patch failed at 0001 Add more foo.
The copy of the patch that failed is found in:
   c:/tmp/git-eol/repo/.git/rebase-apply/patch
When you have resolved this problem, run "git am --resolved".
If you prefer to skip this patch, run "git am --skip" instead.
To restore the original branch and stop patching, run "git am --abort".
$ git am --abort
So the patch doesn't apply well out-of-the-box.  Using --keep-cr was as expected though:
$ git am --keep-cr 0001-Add-more-foo.patch
Applying: Add more foo.
$
Okay.  So let's try this with core.autocrlf=true (in a different repository):
# Removed the initial commands...
$ git format-patch HEAD~1
0001-Add-more-foo.patch
$ git reset --hard HEAD~1
HEAD is now at 525b5aa Initial commit.
$ git am 0001-Add-more-foo.patch
Applying: Add more foo.
$ git config --get core.autocrlf
true
The patch in this case had LF endings everywhere.