The Leicester City man made his first international appearance in the 2-1 loss for Roy Hodgson's men, playing for 84 minutes at the base of the hosts' midfield diamond.
Drinkwater relished his England debut but rued the defeat as he set his sights on continuing to impress Hodgson in order to win a place on the plane to France for the tournament in June.
"Hopefully I have not done myself any harm, it's just about carrying on now until the end of the season and seeing if I can add to it," he said to Sky Sports on his Euro 2016 chances.
"The lads have made it easy for me all week, it's an easy team to settle in to.
"A bit of game time helped me - I was able get on the ball, so it's been easy to settle into the squad.
"I enjoyed it, I saw plenty of the ball, but maybe could have moved it a bit faster. I enjoyed the position.
"Personally I enjoyed my debut - it was a big achievement - but overall I was disappointed with the result."
Jamie Vardy netted England's only goal of the game, following up from his strike in Saturday's win over Germany, and Drinkwater thinks his Leicester team-mate is getting better with every goal he scores.
"He's got two in two now, he's more than capable of doing it," added Drinkwater.
"The more goals he gets the better he is and the more confidence he will get. He's flying and the more he gets on the ball the better."