You need to doubly evaluate the substitution string:

use strict;
use warnings;

my $fileStr = <<'END_FILE';
([A-Za-z])\.[A-Za-z],$1
END_FILE

open my $inFile, "<", \$fileStr or die "Can't open file: $!\n";

my $rrow_test = 'A.BC';

while (my $regex = <$inFile>) {
    chomp $regex;

    my @rex = split (/,/, $regex);

    $rrow_test =~ s/$rex[0]/$rex[1]/eeg;
    print "$rrow_test\n";
}

close $inFile;
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Prints:

AC
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Note that double evaluating the string is like using string eval and is generally considered evil, or at least a huge security risk. For example, try it with the string '([A-Za-z])\.[A-Za-z],@{[print "this could have been: `rm *`\n"]}' in $fileStr