Professor Foias was named Fellow by the Council of the American Association for the Advancement of Science in 1987. The Association is the leading general scientific organization in the United States. Those chosen as Fellows are those whose efforts on behalf of the advancement of science or its applications are scientifically or socially distinguished.

Professor Foias was elected honorary member of the Hungarian Academy of Science in 1993 and of the Romanian Academy in 1994.

In 1995, Foias was awarded the Norbert Wiener Prize in applied Mathematics. The prize is awarded every five years for an outstanding contribution to applied mathematics in the highest and broadest sense. The award is made jointly by the American Mathematical Society (AMS) and the Society for Industrial and Applied Mathematics (SIAM).

Professor Foias was awarded the Rothrock Faculty Teaching Award in 1995. This award recognizes excellence in teaching of mathematics and is given by the Department of Mathematics of Indiana University in Bloomington. In the academic year 1997-98, Foias was the 1998 Ulam Scholar at Los Alamos National Laboratory.