Does having an MSc in a UK university increase chances of being accepted for PhD programme at the same university? I had a conversation with a professor who told me that getting into a PhD programme at some university (in the UK) is very difficult without also having an MSc from that same university. Is this true?

If it's relevant, we were specifically taking about Economics PHDs.
It depends?

I don't think there really is a straightforward answer to this - as there can be some institutional difference.
Though I would in general be a little bit surprised if it were required.

For people who have obtained a UK BSc (it may need to be an honours degree), it is possible to directly progress to a PhD, provided you get a First (maybe also a 2.1) in the BSc. (No idea about a BA - and all related types of Bachelor's degrees.)
This definitely applies to at least 3 universities in the engineering field.

Now: If I am not mistaken, I believe that the majority of people who were "doing a PhD" with a non-UK-education background had a Masters Degree before starting a PhD. (During my time at a UK university.)
However, this wasn't neccessary a Master's Degree from the institution at which these people were working towards a PhD. (Though one person was on some integrated course which was a 1 years Masters followed by a PhD.)

In this context, it may be interesting to note, that the "Student Loans Company" (i.e. the state) will loan the funds for a Bachelor's degree, but not a Masters. A PhD is typically funded (unless one goes for the self funded option). 
This means, that unless you have wealthy parents or an employer paying for it, a Masters Degree in the UK makes little sense if you plan to follow up with a PhD and going Bachelors -> PhD makes more sense.

If we now look at the rest of Europe, it is pretty commonplace to obtain a Masters Degree before commencing a PhD. (Though some countries may have some different routes, such as the old German "Diplom" and the French Engineers that may lead to a PhD/Doctorate.)

And if you want a definite answer, the only method is to check the requirements on the university's website, as only they will be able to give you a definite answer.