Recently, minimal linear codes have been extensively studied due to their applications in secret sharing schemes, secure two-party computations, and so on. Constructing minimal linear codes violating the Ashikhmin-Barg condition and then determining their weight distributions have been interesting in coding theory and cryptography. In this paper, a generic construction for ternary linear codes with dimension $m+2$ is presented, where $m$ is an integer, and a necessary and sufficient condition for this ternary linear code to be minimal is derived. Based on this condition and Krawtchouk Polynomials, a new class of minimal ternary linear codes violating the Ashikhmin-Barg condition are obtained, and then their complete weight enumerators are determined.