I've not done any of this and I'm an EU citizen, but from what I read on the internet, this is true at least for the University of Salzburg (probably all of Austria), IF you immigrate to Austria by using your EC resident status (and not by getting a student visa from a third country, except for those from many african countries), as its website says (note that these are the waiver conditions, the normal fees are indeed higher for foreigners):
Tuition fees are waived temporarily for the length of the prescribed
standard duration of the respective degree program, plus two extra
semesters (tuition-free period) for the following groups of degree
program students:
· Austrian citizens
· Citizens of EU countries
· Citizens of EEA countries (Norway, Iceland, Liechtenstein)
· Swiss citizens
· Refugees under the provisions of the Geneva Convention
(including students from other EU countries)
· Beneficiaries of subsidiary protection status
· Privileged third-country nationals (begünstigte
Drittstaatsangehörige) who hold one of the following types of
residence permits:
EC Long-Term Residence Permit (Daueraufenthalt - EG) issued by the
appropriate Austrian authority
EC Long-Term Residence Permit (Daueraufenthalt - EG) issued by the
appropriate authority of another EU country, plus Austrian residence
permit
Permanent residence permit (Daueraufenthaltskarte) issued by the
appropriate Austrian authority
Students covered by the provisions of the ordinance governing
admission of certain groups of potential students
(Personengruppenverordnung)
Students who are holders of a residence permit other than a Student
Residence Permit (Aufenthaltsbewilligung Studierende)
These students must prove their status to the Admissions Office by
providing all the necessary documents (residence permit,
administrative decision on asylum application, etc.).
See more at http://www.uni-salzburg.at/index.php?id=45122&L=1
It's also worth mentioning that some universities do not make non-EU citizens or foreigners in general pay more at all so you may not even be concerned by this.
Also if your country is on some poor country list, certain universities or countries do not make you pay anything at all.
My tip would be to choose what you want to do, select a few universities and then read their tuition fees rules and the country rules very closely, and contact them to get confirmation.