The Vatreni trailed 1-0 at half time but took control of the match after the break, as Ivan Perisic equalised in the second half and Mario Mandzukic sealed the Three Lions' fate deep into extra time.
Speaking after the match, and quoted by Sky Sports, Dalic claimed: "We were the better team in all segments of the game." He also gave a detailed explanation of how his side's approach had thwarted England.
"We did not underestimate them. We respected them because we analysed them. We knew what they were doing and where," insisted Dalic, explaining: "We played a high press on their back line. We closed down [John] Stones and [Jordan] Henderson. We nullified them."
Dalic also revealed details of what he told his players at half time, after they had struggled to cope with a strong start by England. "During half-time, I told the players 'calmly pass the ball, do not lose your heads'," he said.
His players certainly seemed to follow his instructions, as Croatia dominated possession for much of the remainder of the match.
Dalic was evidently impressed by his charges' performance, as he claimed: "I think this was our best game. We played today better than against Argentina." He was referring to his side's crushing 3-0 victory over La Albiceleste in the group stage.
Croatia's win over England means that they will face France in Sunday's final. The Vatreni have never beaten Les Bleus, who defeated them in the World Cup semi-final in 1998.
As with England, Dalic sounded respectful but not fearful of the French. "We're facing another daunting task but it's going to be a fantastic match," he predicted, adding: "We are ready for it."