Sharma sent back Compton for 3 to leave England 3 for one and then trapped the in-form Cook leg before for 1 to leave the visitors gasping at 16 for 2 in the 11th over.
Jonathan Trott and and Kevin Pietersen, both on 28 and with four fours each, played aggressively to steady the rocking boat and take the visitors to lunch at 61 on a pitch displaying some uneven bounce.
Sharma, who had picked up one wicket in the second Test at Kolkata lost by India to trail by 1-2, bowled with good rhythm to rattle the visiting team after Cook won the toss for the first time in the series and promptly opted to bat first on the dry VCA Stadium track sporting some cracks.
India, who chose to go in with only one fast bowler in Sharma to accommodate the third regular spinner leggie Piyush Chawla, made England struggle for runs with the gangling Delhi pacer, at last showing some aggression in the rubber.
Sharma, who was unwell before the first Test at Ahmedabad and was not picked for the second Test in Mumbai as India went in with three frontline spinners, was not averse to digging the ball in short.
And it was a short, wide ball that accounted for Compton who nervously prodded at it and edged to wicketkeeper Mahendra Singh Dhoni.
It was only the third time that India had managed to break the opening stand between Compton and Cook below 30 and the second successive time in single figures after the second innings in Kolkata.
Trott, lucky not to be given out leg before off Sharma by umpire Kumar Dharmasena, started off with a flowing straight drive off Pragyan Ojha who started the bowling attack with Sharma.
