The large gas and dust reservoirs of submillimeter galaxies (SMGs) could potentially provide ample fuel to trigger an active galactic nucleus (AGN), but previous studies of the AGN fraction in SMGs have been controversial largely due to the inhomogeneity and limited angular resolution of the available submillimeter surveys. Here we set improved constraints on the AGN fraction and X-ray properties of the SMGs with Atacama Large Millimeter/submillimeter Array (ALMA) and Chandra observations in the Extended Chandra Deep Field-South (E-CDF-S). This study is the first among similar works to have unambiguously identified the X-ray counterparts of SMGs; this is accomplished using the fully submillimeter-identified, statistically reliable SMG catalog with 99 SMGs from the ALMA LABOCA E-CDF-S Submillimeter Survey. We found 10 X-ray sources associated with SMGs (median redshift z = 2.3), of which eight were identified as AGNs using several techniques that enable cross-checking. The other two X-ray detected SMGs have levels of X-ray emission that can be plausibly explained by their star formation activity. Six of the eight SMG-AGNs are moderately/highly absorbed, with N H > 1023 cm-2. An analysis of the AGN fraction, taking into account the spatial variation of X-ray sensitivity, yields an AGN fraction of 17^{+16}_{-6}% for AGNs with rest-frame 0.5-8 keV absorption-corrected luminosity >=7.8 × 1042 erg s-1 we provide estimated AGN fractions as a function of X-ray flux and luminosity. ALMA's high angular resolution also enables direct X-ray stacking at the precise positions of SMGs for the first time, and we found four potential SMG-AGNs in our stacking sample.