In Section 5.5.5 of Boyd's book "Convex Optimization", the book states that the primal problem can be solved by the dual problem. More precisely, suppose we have strong duality and an optimal $\{\lambda^{\ast}, \nu^{\ast} \}$ is known. Suppose that the minimizer of $\mathcal{L} (x,\lambda^{\ast}, \nu^{\ast})$, i.e., the solution of
$$\underset{x}{minimize} \; f_0 (x) + \sum_{i=1}^m \lambda_{i}^{\ast} f_i(x) + \sum_{i=1}^p \nu_{i}^{\ast} h_i (x)$$
is unique. (For a convex problem this occurs, for example, $\mathcal{L} (x,\lambda^{\ast}, \nu^{\ast})$ is a strictly convex function of $x$).
My question is that: Because linear/affine functions are not strictly convex, could we get the globally optimal solution to the primal problem by solving the dual problem?
In convex programming under a constraint qualification (slater for example)
The Primal problem has solution if and only if the dual problem has solution.
In this case the optimal value of both problem is same (strong duality holds) and moreover focusing complementary slackness conditions , it makes a bridge between optimal solutions of primal problem to dual problem.
Therefor the answer of your question is "YES" under presence of a constraint qualification.
EDIT
For the particular case, Linear programing , there is no need of constraint qualifications, (in precise words the linear CQ is automatically holds). Also in the later case things become much easier, through an optimal solution which is vertex one can easily get an optimal solution of primal problem which is again vertex.