Abstract: Non-flat unimodular solvable Lie groups do not admit an Einstein metric (I. Dotti, 1982). So, it is natural to ask how close these groups are to being Einstein manifolds. We prove that any non-flat unimodular solvable Lie group with normal derivation admits a left-invariant quasi-Einstein metric if and only if it is standard, has a one-dimensional center, and has a quasi-Einstein nilpotent group as its nilradical. We also establish some necessary conditions for a nilpotent Lie group to be quasi-Einstein. Building on these results, we obtain a classification of unimodular quasi-Einstein solvmanifolds of dimension 6 or less.