After Anthony Gordon saw a goal ruled out with 20 minutes left for Morgan Gibbs-White blocking a free-kick in the build-up, England continued on the front foot and were eventually rewarded during stoppage time when substitute Smith Rowe slotted home following a counter-attack.
England reached the semi-finals of the tournament in 2017, but failed to make it out of the group stage in both 2019 and 2021.
With holders Germany and Israel having played out a 1-1 draw in Kutaisi, Carsley’s side now find themselves in the driving seat for qualification from Group C.
“I am really pleased to get the result,” Carsley said on UEFA.com.