We perform a systematic study in the computational complexity of the connected variant of three related transversal problems: Vertex Cover, Feedback Vertex Set, and Odd Cycle Transversal. Just like their original counterparts, these variants are NP-complete for general graphs. A graph $G$ is $H$-free for some graph $H$ if $G$ contains no induced subgraph isomorphic to $H$. It is known that Connected Vertex Cover is NP-complete even for $H$-free graphs if $H$ contains a claw or a cycle. We show that the two other connected variants also remain NP-complete if $H$ contains a cycle or claw. In the remaining case $H$ is a linear forest. We show that Connected Vertex Cover, Connected Feedback Vertex Set, and Connected Odd Cycle Transversal are polynomial-time solvable for $sP_2$-free graphs for every constant $s\geq 1$. For proving these results we use known results on the price of connectivity for vertex cover, feedback vertex set, and odd cycle transversal. This is the first application of the price of connectivity that results in polynomial-time algorithms.