... $dbh ... is defined in require "incl/common.pl"; without the my keyword.

Then it is a package-global variable. What namespace (package) does incl/common.pl use? If there is no explicit package statement in the required file, I believe it will be in the default main package (update: or whatever package is active in the file into which incl/common.pl is required). So, try
    my $update_time = $main::dbh->selectrow_array("SELECT NOW()");
i.e., use the fully qualified variable name.

Also, $::dbh is shorthand for $main::dbh.