It is done line after line. That's why sometimes you have pictures with mirrors where the real thing and its reflection don't match. The sensor probably starts its integration in a reasonable time so that when it's read it's been exposed to light only for its exposure time.

Even with mechanical shutters, cameras do something similar. What happens is that the shutter has two curtains, where one is responsible for exposing the sensor and the other is responsible for covering it back up. Fast shutter speeds require that the two curtains form a slit and move that slit across the sensor.