We show that the Dzyaloshinskii-Moriya interaction (DMI) can lead to a tilting of the domain wall (DW) surface in perpendicularly magnetized magnetic nanotracks when DW dynamics are driven by an easy axis magnetic field or a spin polarized current. The DW tilting affects the DW dynamics for large DMI, and the tilting relaxation time can be very large as it scales with the square of the track width. The results are well explained by an extended collective coordinate model where DMI and DW tilting are included. We propose a simple way to estimate the DMI in magnetic multilayers by measuring the dependence of the DW tilt angle on a transverse static magnetic field. These results shed light on the current induced DW tilting observed recently in Co/Ni multilayers with structural inversion asymmetry.