A projective variety $V$ is $\left(1,h\right)$-defective if the Grassmannian of lines contained in the span of $h+1$ independent points on $V$ has dimension less than the expected one. In the present paper, which is inspired by classical work of Alessandro Terracini, we prove a criterion of $\left(1,h\right)$-defectivity for algebraic surfaces and we discuss its applications to Veronese embeddings and to rational normal scrolls.