The zero forcing number of a simple graph, written $Z(G)$, is a NP-hard graph invariant which is the result of the zero forcing color change rule. This graph invariant has been heavily studied by linear algebraists, physicists, and graph theorist. It's broad applicability and interesting combinatorial properties have attracted the attention of many researchers. Of particular interest, is that of bounding the zero forcing number from above. In this paper we show a surprising relation between the zero forcing number of a graph and the independence number of a graph, denoted $α(G)$. Our main theorem states that if $G \ne K_4$ is a connected, cubic, claw-free graph, then $Z(G) \le α(G) + 1$. This improves on best known upper bounds for $Z(G)$, as well as known lower bounds on $α(G)$. As a consequence of this result, if $G \ne K_4$ is a connected, cubic, claw-free graph with order $n$, then $Z(G) \le \frac{2}{5}n + 1$. Additionally, under the hypothesis of our main theorem, we further show $Z(G) \le α'(G)$, where $α'(G)$ denotes the matching number of $G$.