Inter Milan midfielder Nicolo Barella headed in the game’s only goal on half-time as the Azzurri moved into top spot in their League A group ahead of the Dutch who won their opening game against Poland.
Italy dominated the contest and arguably should’ve won by more with 18-11 shots and 58-42% possession.
“We should’ve killed the game earlier,” Mancini told RAI Sport.
“I am satisfied with the mentality, the performance in Amsterdam and the lads were extraordinary this evening.”
Elsewhere, Poland came from behind to defeat Bosnia-Herzegovina 2-1 in Zenica in the other Group 1 game.
In League B on Monday, Erling Haaland stole the show with two goals and an assist as Norway thrashed Northern Ireland 5-1 in Belfast.
Four goals were scored inside the opening 20 minutes of a frantic clash but Haaland took the game by the scruff of the neck, with an assist and goal in the second half settling the match.
Romania claimed an impressive 3-2 away win over Austria, marking new manager Mirel Radoi’s first win.
Scotland moved into top spot in Group B2 with a come-from-behind 2-1 win over an undermanned Czech Republic, while Israel and Slovakia drew 1-1.
“We’re always on a hiding to nothing,” Scotland boss Steve Clarke said.
“Everyone told us before the start of the game we had to win the game, so we won the game. We take the three points and go home.”