Some viruses counter host-cell efforts to digest invading viral RNA by using special structures resistant to host RNases, known as exoribonuclease-resistant RNAs (xrRNAs). xrRNAs typically form an unusual fold with the 5'-end threaded through a ring consisting of a multihelix junction closed by a pseudoknot. By using single-molecule force spectroscopy (SMFS), we previously showed that a Zika virus xrRNA is extremely rigid mechanically, withstanding very high forces, and that this mechanical resistance-not simply the knot-like fold topology-is essential for RNase resistance. Here, we have determined which interactions are most important for generating mechanical rigidity in the Zika virus xrRNA, by systematically mutating tertiary contacts. We found that removing any of the tertiary contacts involving the threaded 5' end was sufficient to abrogate mechanical resistance. In contrast, breaking a single pseudoknot base pair was not sufficient to do so: Two broken pairs were needed. This hierarchy of interaction importance for mechanical rigidity was supported by simulations mapping how mechanical tension was distributed within the xrRNA. For all mutants, RNase resistance varied in lock-step with mechanical resistance, confirming the primary role of mechanical rigidity in xrRNA function. This work reveals which interactions are most important for Zika xrRNA function, with implications for targeting the xrRNA therapeutically.