use IO::Compress::Zip qw(zip $ZipError);

my $zip = zip [$0] => 'test.zip', 
Name => 'test.txt', 
exTime => [$atime, $mtime, $ctime] or die;
[download]

Well, it still does not retain the original modified date of the file, losing the seconds resolution. Not sure if this is correct, perhaps someone could share and prove that the original modified date can be retained.
Keeping things simple, Perl shell execution Zip doesn't require much setting to retain the original modified date.

Thanks for the result, but this is what I am getting,
first is the shell zip, which retained the timestamp.
next is perl zip which timestamp became current.
lastly is perl zip with extime which timestamp rounded down to even seconds.

OS RHEL


$ stat abc | grep Mod
Modify: 2017-01-09 18:46:05.000000000 -0800

$ zip abc.zip abc
  adding: abc (stored 0%)
$ unzip abc.zip
Archive:  abc.zip
replace abc? [y]es, [n]o, [A]ll, [N]one, [r]ename: r
new name: abc2
 extracting: abc2
$ stat abc2 | grep Mod
Modify: 2017-01-09 18:46:05.000000000 -0800

$ rm abc.zip

$  perl -M'IO::Compress::Zip qw(zip)' -e ' zip "abc" => "abc.zip" '
$ unzip abc.zip
Archive:  abc.zip
replace abc? [y]es, [n]o, [A]ll, [N]one, [r]ename: r
new name: abc3
  inflating: abc3
$ stat abc3 | grep Mod
Modify: 2017-01-10 15:54:35.000000000 -0800

$ rm abc.zip

$  perl -M'IO::Compress::Zip qw(zip)' -e ' zip "abc" => "abc.zip", exT
+ime => [$atime, $mtime, $ctime] '
$ unzip abc.zip
Archive:  abc.zip
replace abc? [y]es, [n]o, [A]ll, [N]one, [r]ename: r
new name: abc4
  inflating: abc4
$ stat abc4 | grep Mod
Modify: 2017-01-09 18:46:04.000000000 -0800
[download]

Archive::Zip is also not included in the Perl core if that's what you mean. But also see Yes, even you can use CPAN on how you can install modules yourself.