Your code will also match a file named pretxt.
my $exten = "(\.txt|\.csv)";
should be
my $exten = "(\\\.txt|\\\.csv)";  # (\.txt|\.csv)

There's no reason to force the user to specify the parens.
Why did you remove the case-insensitivity?
You also forgot to anchor the regexp.

use strict;
use warnings;

my $exten = "\\\.txt|\\\.csv";  #  \.txt|\.csv

my @files = qw(
   one.txt two.html three.csv four.log pretxt file.txt~ File.Txt
);

foreach (@files) {
  if (/($exten)\z/i) {
    print $_, "\n";
  }
}
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Update: Or if you want to build $exten:

my @exten = qw( .txt .csv );
my $exten = join('|', map quotemeta, @exten);
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