For our internal Wiki (don't know which Mediawiki version, but oldish), I use MediaWiki::API, which can work from a username/password.

The relevant parts look like:

...
my ($url, $user, $passwd) = qw(http://wiki.serv.er/api.php rob.otter T
+0pSeCrEt);
...
use MediaWiki::API;
my $mw = MediaWiki::API->new();
$mw->{config}->{api_url} = $url;

# log in to the wiki
$mw->login( { lgname => $user, lgpassword => $passwd } )
  || die $mw->{error}->{code} . ': ' . $mw->{error}->{details};
...

    my $pagename = ...;
    
    my $ref = $mw->get_page( { title => $pagename } );
    unless ( $ref->{missing} ) {
        my $timestamp = $ref->{timestamp};
        my %p = (
            action => 'edit',
            basetimestamp => $timestamp, # to avoid edit conflicts
            title => $pagename,
            bot => '1',
            nocreate => '1',
            text => '... this is the text ...',
        );
        $mw->edit(\%p)
          || die $mw->{error}->{code} . ': ' . $mw->{error}->{details}
+;
    };

...
[download]

The content of the %p hash are documented on the corresponding api.php page, e.g. https://en.wikipedia.org/w/api.php?action=help&modules=edit.