Brook and Livingstone's fifth-wicket stand helped the defending champions get within seven runs of the target, creating a clearer path to the semis despite hosts West Indies pulling ahead in the group with a dominant nine-wicket win over the United States.
"I think Brook and Livingstone had a fantastic partnership on a slow wicket where other guys struggled," Buttler told reporters in Gros Islet. "The big picture is obviously net run rate."
West Indies, tied with England on two points, rose to second place in Group Two thanks to a superior net run rate of 1.814.
Third-placed England have a net run rate of 0.412, close to leaders South Africa (0.625) who have four points.
This makes it easier for England to overtake South Africa's net run rate with a win over the USA on Sunday if the leaders lose to West Indies. If South Africa wins, England would ensure second place outright with four points.
South Africa, who have won all six matches in the tournament, now face a potential do-or-die scenario where a loss to West Indies may leave them out of the top two and a place in the semis.