Provide functionality from newer system

I have a question about cross-linking or embedding or if there is a more suitable option.

The Requirement

So we're developing a web app with new functionality, but our older system is still widely used. Now our boss decided that it would be a good idea if the old system could also benefit from the new funtionality, but without the need of implementing it twice. This also includes the UI, which is a bit of a problem for me because the systems look and feel entirely differently.

My approaches

I'm trying to find a way to make certain pages of the new web app usable in the old web app, with as little irritation to the user as possible. So far, I only came up with the "classical" approaches of a) embedding the site or b) placing a link which opens the site in a new tab.

My questions

Is there another, probably better approach? Is it recommended to inform the users about what is going to happen when clicking on a link that leads them somewhere else? What works best with user expectation, as I don't want to throw them off.

(If it is relevant: people use our systems in a professional context and they have to work with our software. The thing I'm worried about is that users might be irritated and lose focus on their task.)