Antonio Conte's men produced a strong performance in the first half and were deserving of the lead, which was given to them by the lively Willian.
But Romelu Lukaku levelled in the 39th minute with a well-worked goal and United ultimately secured victory through substitute Jesse Lingard in the second period.
Drinkwater, who came through United's youth system, believes a defeat was harsh on Chelsea, who now find themselves trailing Jose Mourinho's men by six points in the Premier League.
"We definitely should have got something from the game," Drinkwater told Chelsea's official website.
"In the first half we were very dominant. In the second half it was a bit different, it was more even, but we didn't deserve to lose that game, no chance.
"I don't know the stats, but the first half felt good. Their goal came out of nowhere. We had a bit of confidence coming into the second half, but it didn't go our way.
"It was more of a battle which suited them and took the game away from us.
"Football goes against you sometimes and it was one of those days. There are obviously bits we could have improved on, but we have to got to look forward and on to the next game now.
"It's down to us. We have got to get our performances and our results in order and hopefully our position will look after itself."