Cook, who was the hero of England's 2-1 series win with 568 runs in all, was given out caught behind on the fourth day of the fourth Test in Nagpur despite replays suggesting no contact between bat and ball.
The opener was dismissed lbw in the first innings but replays suggested that the ball was missing leg stump.
"We made our views pretty clear at the beginning of the series. We believe DRS improves decision-making on the field. That has got to be good for the game. It is disappointing we are not using it in this series - and we should be," Morris told 'BBC Radio Five Live's Sportsweek' programme.
"Over the last few years the DRS has really bedded into our game, has improved our game. We should be using it," he said.
Reflecting on the team's performance, Morris lauded Cook, who stood out with his superb batting.
"Alastair, for the past 15 to 18 months, has been captain of the one-day squad and has really taken that team forward. The way he has gone about his business in India has been hugely impressive with the bat and as a leader," he said.
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"He has got the respect of the changing room. On his first major tour, he has led the team extremely well."
Morris also praised Kevin Pietersen, who has been reintegrated into the squad after a string of dressing-room controversies.
"It has worked out really well both on and off the field. We've moved on from where we were in the summer. Kevin sat down with the players and the management team and any outstanding issues have been put to bed," he said.
"Off the field, he has done everything we've asked of him and on the field he has made huge contributions. The innings in Mumbai was a great innings, one of the great modern Test innings by an England batsman," he added.