grep -il "^$input" *
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is equivalent to the following Perl one-liner in bash:

perl -lne '$seen{$ARGV} = 1 if /^'"$input/i }{ print for keys %seen" *
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or the following script:

#!/usr/bin/perl
use strict;
use warnings;

my $input = shift;
for my $file (glob '*') {
    open my $IN, '<', $file or die $!;
    while (<$IN>) {
        if (/^$input/i) {
            print "$file\n";
            last;
        }
    }
}
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Run as script.pl "$input".

For example with a combination of readdir and grep:

my $dirname= '/some/directory/with/files';
my $input= '2013-09-04';
opendir my $dh, $dirname
    or die "Couldn't read '$dirname': $!";
my @found= grep /^\Q$input\E/, readdir $dh;

...
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In my opinion the easiest way is to use File::Glob::bsd_glob:

use File::Glob qw( bsd_glob );
my @found= bsd_glob("$input*");
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That's not what grep -il is doing, it outputs the filenames of the files whose content contains the string. So IMHO it's easier to use grep, unless there's more to this problem that we know about.