my $sqlstatement=("UPDATE servers SET nicl_driver_version = '$driver' 
+WHERE '$field1' = '$servlist'");
[download]

you also don't appear to be using $field2 or $vals for anything, and don't need the use Win32::ODBC; line for anything as well (you are not using that module in your code at all).

One last thing is that you might want to prepare your SQL statement outside your loop, and then execute it on the inside, so you don't have to prepare the same statement time and time again unnecessarily). So your code might look something like this(untested):

use DBI;
use strict;

my $DSN='Database';
my $field1 = 'Server';

my $dbh=DBI->connect("dbi:ODBC:$DSN", "admin", "") 
   or die "$DBI::errstr\n";

my $sth = $dbh->prepare(
   "UPDATE Servers 
    SET nic1_driver_version= ? 
    WHERE '$field1' = ?" );

#Open file for reading
open(INPUT,"file.txt") || die;
while (<INPUT>) {
 chomp;
 my ($servlist, $driver)=split(/,/);
 $sth->execute($driver,$servlist) or die;
}
close (INPUT);
$dbh->disconnect;
[download]

-enlil