We have compiled a sample of 14 of the optically brightest radio-quiet quasars ($m_{i}$~$\le$~17.5 and $z$~$\ge$~1.9) in the Sloan Digital Sky Survey Data Release 5 quasar catalog that have C IV mini-BALs present in their spectra. X-ray data for 12 of the objects were obtained via a Chandra snapshot survey using ACIS-S, while data for the other two quasars were obtained from archival XMM-Newton observations. Joint X-ray spectral analysis shows the mini-BAL quasars have a similar average power-law photon index ($\Gamma\approx1.9$) and level of intrinsic absorption ($N_H \lesssim 8\times 10^{21} \ {\rm cm}^{-2}$) as non-BMB (neither BAL nor mini-BAL) quasars. Mini-BAL quasars are more similar to non-BMB quasars than to BAL quasars in their distribution of relative X-ray brightness (assessed with $\Delta\alpha_{\rm ox}$). Relative colors indicate mild dust reddening in the optical spectra of mini-BAL quasars. Significant correlations between $\Delta\alpha_{\rm ox}$ and UV absorption properties are confirmed for a sample of 56 sources combining mini-BAL and BAL quasars with high signal-to-noise ratio rest-frame UV spectra, which generally supports models in which X-ray absorption is important in enabling driving of the UV absorption-line wind. We also propose alternative parametrizations of the UV absorption properties of mini-BAL and BAL quasars, which may better describe the broad absorption troughs in some respects.