England’s win in Antigua left them knowing they would progress as long as Australia beat Scotland.
Scotland played superbly but Australia, who had already secured their place in the Super 8s, chased down 181 with two balls to spare.
It means England progress to a group with the United States, South Africa and co-hosts West Indies, who they play in their next match at 01:30 BST on Thursday.
“It was a nervous watch at times but thankfully we can now prepare for a big week of crucial matches in St Lucia and Barbados against strong opposition,” said Buttler.
As the rain fell in Antigua, England were 46 minutes away from the match being abandoned – a result that would have knocked them out regardless of how Scotland fared.
Buttler said his side “reacted really well” to the situation and praised the calming influence of former West Indies all-rounder Kieron Pollard, who is part of the England coaching staff for the tournament.
“He’s been around and seen it all,” Butter said of the two-time World Cup winner.
“He has that sort of laid-back attitude of ‘this is just what happens, we can deal with it when it comes’.”
England travelled to St Lucia on Sunday but are waiting for a clearer picture on the fitness of Liam Livingstone, who injured his side while batting against Namibia.