A subset \(X\) of a topological group \(G\) is said to be a topological generating set for \(G\) if the smallest closed subgroup containing \(X\) is \(G\) itself, or, equivalently, the group generated by \(X\) is dense in \(G\). Therefore and in this paper, the authors define the topological generating rank \(d(G)\) of a connected Lie group \(G\) as the minimal cardinality of a topological generating set of \(G\). \textit{K. H. Hofmann} and \textit{S. A. Morris} [Semin. Sophus Lie 2, No. 2, 123--134 (1992; Zbl 0788.22004)] point out: An analysis of the topological rank of a connected Lie group requires an answer to the following question: What is the topological generating rank of a solvable connected Lie group? The aim of this paper is to give a complete answer to this question. If \(G\) is a connected solvable Lie group then they reduce the question to the case that \(G\) belongs to a class of connected Lie groups such that 1. \(G\) is metabelian, 2. The toplogical commutator subgroup \(A:=\overline{G'}\) is a vector group, i.e. isomorphic to \(\mathbb R^n\) for some \(n\), 3. \(A\) contains no non-trivial vector fixed by \(G/A\). 4. The natural representation of \(Q\) on \(A\) is semisimple. Then \(d (G)\) is the maximum of the following two numbers: \(d (Q)\) and one plus the maximum of the multiplicities of the non-trivial isotypic components of the \(\mathbb RQ\)-module \(A\).