Writing a reverse regex seems like too much work for a lazy Perl hacker. I think you're on the right track, especially with the end-of-line anchor:

(my $ext = $url) =~ m!\.([^/]+)$!;

With greedy quantifiers like + and *, the regex engine parses from right to left anyway. So take advantage of that.

Somehow, that construct isn't behaving for me after a long week, so here's another approach:

#!/usr/bin/perl -w

use strict;

print "Url: ";
chomp(my $url = <STDIN>);

if ($url =~ m!\.([^/]+)$!) {
    print "Ext: $1\n";
} else {
    print "Not found.\n";
}
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