Using DNS is not a redirect. You simply pointed another domain name to the same server. Nothing more. There is serious danger in this in that you will/can end up with duplicate content that will harm SERP performance severely.
TO avoid this, traditionally, you would want to have both domains configured on a web server (it does not have to be the same server) and do a 301 redirect from the old site to the new site. Often this is done with a blanket redirect from the old companies site to the new site and then redirects for particular pages one at a time on the new site to match the URI from the old site to point to the appropriate page on the new site. If you do not have a matching page from the old site to the new site, it is okay to let it return a 404 Not Found error though if you can, a 410 Gone error is better but may be more work that you want to do. This way, you avoid the duplicate content issue.
I would suggest doing this quickly since you have already allowed the search engines spider both sites. You will want to make the changes quickly to avoid as much loss of SERP performance as possible. This can be done with an audit of both sites and not worrying about pages that do not matter. Just focus on the ones that do matter.
[Update]
Here is an example of a blanket redirect:
RewriteCond %{HTTP_HOST} ^oldsite\.com$ [NC,OR]
RewriteCond %{HTTP_HOST} ^www\.oldsite\.com$ [NC]
RewriteRule ^(.*)$ http://www.newsite.com/$1 [R=301,L]
I would be putting this on the old site to redirect the entire thing over to the new site.
Here is an example of a specific page redirect:
Redirect permanent /about/aboutus.html http://www.newsite.com/aboutus/
I would be putting this on the new site to redirect specific pages one at a time. Of course, you can redirect entire directories or do not need a redirect of the URI (path/pages) are identical.
