By adapting a new mathematical approach to the problem of steady free-surface Euler flows with surface tension recently devised by the present author, it is demonstrated that exact solutions for steady, free-surface multipole-driven Hele-Shaw flows with surface tension can be constructed using similar methods. Moreover, a (one-way) mathematical transformation between exact solutions to the two distinct free-boundary problems is identified: known exact solutions for free-surface Euler flows with surface tension are shown to automatically generate steady quadrupolar-driven Hele-Shaw flows (with non-zero surface tension) existing in exactly the same domain with the same free surface. This correspondence highlights the essential dynamical differences between the two physical problems. Using the transformation, the exact Hele-Shaw analogues of all known exact solutions for free-surface Euler flows (including Crapper's classic capillary water wave solution) are catalogued thereby producing many previously unknown exact solutions for steady Hele-Shaw flows with capillarity. In particular, this paper reports what are believed to be the first known exact solutions for Hele-Shaw flows with surface tension in a doubly-connected fluid region.