This follows from Theorem 9.3 (page 216) of the book "Compact tranformation groups" by Bredon (note that it is freely available online).
More is true: Any effective group action of a torus on a torus is free. The proof starts along the lines you mentioned via fundamental group considerations.
The result is originally due to Connor and Montgomery:
Conner, P. E., and Montgomery, DC. Transformation groups on a K(n, l), I. Michigan Math. J. 6 (1959), 405-412.