We report exploratory Chandra observations of 14 high-redshift (z=4.06-5.27), optically selected quasars. Ten of these quasars are detected, increasing the number of z>4 X-ray detected quasars by 71%. Our detections include four of the five highest-redshift X-ray detected quasars to date, among them SDSSp J021043.17-001818.4, the highest-redshift (z=4.77) radio-loud quasar detected in the X-ray band. The four undetected objects are the Broad Absorption Line quasars SDSSp J112956.10-014212.4 and SDSSp 160501.21-011220.0, the weak emission-line quasar SDSSp J153259.96-003944.1, and the quasar PSS 1435+3057. A comparison of the quasars spectral energy distributions (by means of the optical-to-X-ray spectral index, alpha_ox) with those of lower-redshift samples indicates that the Chandra quasars are X-ray fainter by a factor of approx 2. X-ray faintness could be associated with the presence of large amounts of gas in the primeval galaxies harboring these high-redshift quasars, as suggested by recent studies conducted on z>4 quasars in other bands. Using the current Chandra data, predictions for the next generation of X-ray observatories, Constellation-X and Xeus, are also provided.