We present results from a 25 ksec CHANDRA ACIS-S observation of galaxies NGC 4438 and NGC 4435 in the Virgo Cluster . X-ray emission in NGC 4438 is observed in a \sim 700 pc nuclear region , a \sim 2.3 kpc spherical bulge , and a network of filaments extending 4 - 10 kpc to the west and southwest of the galaxy . The X-ray emission in all three regions is highly correlated to similar features observed in H \alpha . Spectra of the filaments and bulge are well represented by a 0.4 keV MEKAL model with combined 0.3 - 2 keV intrinsic luminosity L _ { X } = 1.24 \times 10 ^ { 40 } erg s ^ { -1 } , electron densities \sim 0.02 - 0.04 cm ^ { -3 } , cooling times of 400 - 700 Myr and X-ray gas mass \lesssim 3.7 \times 10 ^ { 8 } M _ { \odot } . In the nuclear region of NGC 4438 X-ray emission is seen from the nucleus and from two outflow bubbles extending 360 pc ( 730 pc ) to the northwest ( southeast ) of the nucleus . The spectrum of the northwest outflow bubble plus nucleus is well fitted by an absorbed ( n _ { H } = 1.9 ^ { +1.0 } _ { -0.4 } \times 10 ^ { 21 } cm ^ { -2 } ) 0.58 ^ { +0.04 } _ { -0.10 } keV MEKAL plasma model plus a heavily absorbed ( n _ { H } = 2.9 ^ { +3.1 } _ { -2.0 } \times 10 ^ { 22 } cm ^ { -2 } ) \Gamma = 2 , power law component . The electron density , cooling time , and X-ray gas mass in the northwest outflow are \sim 0.5 cm ^ { -3 } , 30 Myr and 3.5 \times 10 ^ { 6 } M _ { \odot } . Weak X-ray emission is observed in the central region of NGC 4435 with the peak of the hard emission coincident with the galaxy ’ s optical center ; while the peak of the soft X-ray emission is displaced 316 pc to the northeast . The spectrum of NGC 4435 is well fitted by a non-thermal power law plus a thermal component from 0.2 - 0.3 keV diffuse interstellar medium gas . We argue that the X-ray properties of gas outside the nuclear region in NGC 4438 and in NGC 4435 favor a high velocity , off-center collision between these galaxies \sim 100 Myr ago ; while the nuclear X-ray emitting outflow gas in NGC 4438 has been heated only recently ( within \sim 1 - 2 Myr ) by shocks ( v _ { s } \sim 600 km s ^ { -1 } ) possibly powered by a central active galactic nucleus .