A 1-0 win over Poland in Tiblisi on Saturday secured progression to the final round of qualifying as Group 3 winners - following a 3-0 win over Sweden and 1-1 draw with the hosts Georgia.
"We made a good start to the season with some tough games and I think that helped us out here," said Peacock.
"But those games came in friendly tournaments: this is different. You have to qualify for the next stage of the Euros and that brings with it a new kind of pressure for the boys and they can be proud of themselves for what they've achieved."
England won the four-team tournament - co-hosted at Glanford Park, Sincil Bank and Blundell Park - ahead of Australia, Portugal and Turkey.
Their one outing in North Lincolnshire was a 3-1 victory over Portugal, a result which secured them the trophy.