Let T₁ ∈ B(H₁) be a completely non-unitary contraction having a non-zero characteristic function Θ₁ which is a 2 × 1 column vector of functions in H^∞. As it is well-known, such a function Θ₁ can be written as Θ₁ = w₁m₁[^a1 _b1] where w₁,m₁,a₁,b₁ ∈ H ^∞ are such that W₁ is an outer function with |w ₁| ≤ 1, m₁ is an inner function, |a₁| ² + |b₁|² = 1, and a₁ = 1 (here ∧ stands for the greatest common inner divisor). Now consider a second completely non-unitary contraction T₂ ∈ B(H₂) having also a 2 × 1 characteristic function Θ₂ = w ₂m₂[^a2 _b2]. We prove that T ₂ is quasi-similar to T₂ if, and only if, the following conditions hold: 1. m₁ = m₂, 2. {z ∈ : |w ₁(z)| < 1} = {z ∈ : |w₂(z)| < 1} a.e., and 3. the ideal generated by a₁ and b₁ in the Smirnov class N⁺ equals the corresponding ideal generated by a₂ and b₂.