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In this paper, by using the new concept of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>ϱ</mi><mo>,</mo><mi>ψ</mi><mo>,</mo><mi>ω</mi><mo>)</mo></mrow></semantics></math></inline-formula>-quasiinvexity associated with interval-valued path-independent curvilinear integral functionals, we establish some duality results for a new class of multiobjective variational control problems with interval-valued components. More concretely, we formulate and prove weak, strong, and converse duality theorems under <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>ϱ</mi><mo>,</mo><mi>ψ</mi><mo>,</mo><mi>ω</mi><mo>)</mo></mrow></semantics></math></inline-formula>-quasiinvexity hypotheses for the considered class of optimization problems.