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In this paper, we explore the concept of topologically dense injectivity of monoid acts. It is shown that topologically dense injective acts constitute a class strictly larger than the class of ordinary injective ones. We determine a number of acts satisfying topologically dense injectivity. Specifically, any strongly divisible as well as strongly torsion free S-act over a monoid S is topologically dense injective if and only if S is a left reversible monoid. Furthermore, we establish a counterpart of the Skornjakov criterion and also identify a class of acts satisfying the Baer criterion for topologically dense injectivity. Lastly, some homological classifications for monoids by means of this type of injectivity of monoid acts are also provided.