"If I'm trying to do an exact match..."

Exact match for what?
You didn't specify what $var contains.

Anyway, does this do what you want?

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

my $var = 'Sensor30';

while (<DATA>) {
    chomp;
    print if m/^\Q$var\E$/;
}

__DATA__
>Sensor30
>FooSensor30
> Sensor30
Sensor300
Sensor30
>Sensor3
[download]

Cheers,
Darren