We present a Chandra Director's Discretionary Time observation of PSR J1119-6127 and its compact X-ray pulsar wind nebula (PWN) obtained on 27 October 2016, three months after the Fermi and Swift detection of millisecond bursts in hard X-rays, accompanied by 160 times increase in flux. This magnetar-like activity, the first observed from a rotation-powered radio pulsar, provides an important probe of the physical processes that differentiate radio pulsars from magnetars. The post-burst X-ray spectrum of the pulsar can be described by a single powerlaw model with a photon index of 2.0±0.2 and an unabsorbed flux of 5.7−12 ergs cm −2 s −1 in the 0.5-7.0 keV energy range. At the time of Chandra observations, the pulsar was still brighter by a factor of ∼22 in comparison with its quiescence. The X-ray images reveal a nebula brighter than in the pre-burst Chandra observations (from 2002 and 2004), with an unabsorbed flux of 2.2 +1.1 −0.9 ×10 −13 ergs cm −2 s −1 . This implies a current X-ray efficiency of ≈0.001 at a distance of 8.4 kpc. In addition, a faint torus-like structure is visible along the southeast-northwest direction and a jet-like feature perpendicular to the torus towards the southwest. The PWN is best fitted by an absorbed powerlaw with a photon index of 2.2±0.5 (post-burst). While the pulsar can still be energetically powered by rotation, the observed changes in PSR J1119-6127 and its PWN following the magnetar-like bursts point to an additional source of energy powered by its high-magnetic field.