India dismissed England for 191 shortly before tea, but England finally gave their supporters something to shout about by ending day three on 111-0.
England had struggled in the morning session in Ahmedabad as they quickly went from 69-3 to 110-7.
Matt Prior, Tim Bresnan and Stuart Broad mounted some resistance after lunch but they could not prevent the India bowling attack picking up the final three wickets to dismiss England for 191 with Pragyan Ojha taking 5-45.
India soon asked England to follow on as they eyed an opportunity to decimate England's top order like they did in the first innings.
Cook batted aggressively for a knock of 74 while Compton supported his captain efficiently with a timely 34.
England still have a mountain to climb if they are to even force India to bat again, but an unbeaten century stand provides them with at least some hope when they begin day four tomorrow.