We study the contact 1-Lefschetz condition on compact contact solvmanifolds with an invariant contact form, as introduced by B. Cappelletti-Montano, A. De Nicola and I. Yudin. We prove that the 1-Lefschetz condition on Lie algebras is preserved via 1-dimensional central extensions by a symplectic cocycle, thereby establishing that a unimodular symplectic Lie algebra (h,ω) is 1-Lefschetz if and only if its contactization (g,η) is 1-Lefschetz. We achieve this equivalence by showing an explicit relation between the relevant cohomology degrees of h and g, and also between the commutators [h,h] and [g,g]. By specializing to the nilpotent setting, we prove that 1-Lefschetz contact nilmanifolds equipped with an invariant contact form are quotients of a Heisenberg group by a lattice, and deduce that there are many examples of compact K-contact solvmanifolds not admitting compatible Sasakian structures. Lastly, we construct new examples of completely solvable 1-Lefschetz solvmanifolds, some having the 2Lefschetz property and some failing it.