Try this:

my @previous;
while (<LINE>) {
    if ($_ eq "STRING TO COMPARE WITH\n") {
        print $previous[0];
        last;
    }
    push @previous, $_;
    shift @previous if @previous > 5;
}
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Update: Thanks to jeroenes for spotting a couple of things that could be done better - I've incorporated your suggestions into the code.

Update 2: Thanks to davorg for noticing that print @previous prints out all of the last 5 lines, but pat asked for the line 5 lines previous to the one being searched. I've changed the print @previous line to read print $previous[0] instead.