The Croatians battled back twice to level the scores at 2-2 on the stroke of halftime, but were unable to manage a reply to two England goals in the second half.
Dalic, who led his country to the 2018 World Cup final and the semi-finals four years ago in Qatar, said it was vital his players rallied quickly for their remaining group matches against Panama and Ghana.
"It's not easy to lose," he told reporters. "You lose your self-confidence, and that's not good.
"We must prepare quite well for the next two games in the first stage. It's very important for us to be at the right level after this defeat. We do not have the right to any more mistakes.
"We do not have the right to bad matches or mistakes. I wouldn't say that this was a bad match. There were some errors we committed, some mistakes we made. We were punished by our opponent. Two matches are ahead of us, new beginnings."
Dalic was particularly scathing of his team's efforts at defending set pieces after two of England's goals followed corners.
"We used to be quite good at set pieces. We didn't concede goals that easily, and we practised for this opponent. We knew the way England were going to play," he said.
"It was disastrous, specifically when it comes to a team like England. About 40% of their goals are from set pieces.
"We must correct it and must avoid similar mistakes in the future," he added.
Dalic said the team had gained confidence from the goals from Petar Musa and Martin Baturina that cancelled out Harry Kane's first two strikes for England.
The goal that England's Jude Bellingham scored soon after halftime had knocked the stuffing out of his players, however.
"We had 10 minutes after that, which were really quite bad," he said.
"We tried to get back into the game, but we didn't succeed. It was a strange match. We made a lot of mistakes, specifically when it came to the set pieces, but I must congratulate England."