The authors consider the perturbed evolution inclusion \[ -\frac{du}{dr}\in A(t)u(t)+f(t,u(t)) \tag{1} \] in a separable Hilbert space \(H\), \(du\) is the Stieltjes or differential measure, \(dr\) is a positive measure, \(A(t):D(A(t))\rightarrow2^{H}\) is a maximal monotone operator for every \(t\in I\equiv[0,T]\) and \(f:I\times H\rightarrow H\) is a Carathéodory mapping that satisfies a Lipschitz-type condition and a linear growth condition. Theorem 3.1 provides existence and uniqueness of a bounded, continuous solution with bounded variation under the assumption that the dependence of \(A\) on \(t\) is of continuous bounded variation dominated by \(dr\). Theorem 3.3 proves existence and uniqueness of a Lipschitz solution for the problem \[ -\frac{du}{dt}\in A(t)u(t)+f(t,u(t))\tag{1} \] under the assumption that \(A\) is Lipschitz. These results are carefully applied to a number of varied applications in the last section: nonemptiness and compactness of the solution set for functional differential inclusions, an existence and uniqueness result for second order evolution inclusions, nonemptiness and compactness of the set of Lipschitz solutions to differential inclusions with convex weakly compact valued upper semicontinuous perturbations, Bolza and relaxation problems in optimal control theory and a Skorokhod problem governed by a maximal montone operator. The authors note that some results involving a sweeping process (such as [\textit{S. Adly} et al., Math. Program. 148, No. 1--2 (B), 5--47 (2014; Zbl 1308.49013)] and some results for subdifferential evolution problems are covered as special cases. The proofs make use of some lemmas from [\textit{M. Kunze} and the third author, Set-Valued Anal. 5, No. 1, 57--72 (1997; Zbl 0880.34017)].