Once a page is out there, Google is just going to crawl it, suck it up, and retain it in its vast and very long term memory.
If the page you want removed is in Google's SERPs, then the noindex alone won't resolve this, it'll simply advise new search engines not to rank it. You need to add nofollow,noarchive to that page's robots command (it can all be in one command) to flush it out of Google etc search cache, and stop the crawler following the links on that page.
Make sure you're allowing search engines to crawl that page via your robots.txt file, otherwise they'll never see this request.
Once your page has gone from the SERPs (do a few site:[yourdomain] queries in Google search box to check), then you can use the robots.txt to advise search not to go there. I'd opt to also keep the noindex,nofollow,noarchive page-level command in the page too, to keep it out of SERPs just in case the robots.txt file gets an edit.
It can take a while to fix this, but it'll get there.