You want to find yarn (not thread) that is similar in color and weight to that in the sweater. Split plies if you have to to get it fine enough. Use a yarn needle (it is fatter than a thread needle, and blunt rather than sharp). Then just try to duplicate the path that the yarn travels in a row, starting off to the side in the undamaged section: up from behind in a lower loop, down thru the front of the corresponding upper loop, up from behind in the next upper loop, back down into the same lower loop you started in, up thru the next lower loop, ...
Keep going until you are into the undamaged section on the opposite side, and repeat on other rows as needed.