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West Indies have won only one Test series since 2013 and have been dogged by financial disputes, but both men and women have reached their T20 finals and the U19s won the World Cup in February.
"We came on a mission inspired by the under-19s and the ladies," said Sammy.
"Now we have two West Indies teams in the final and I know that means a lot to the Caribbean."
As part of financial disputes with the West Indies Cricket Board, players pulled out of a tour of India in 2014 and only signed contracts to take part in the World Twenty20 a month before it was due to start.
With many of their top players opting to play in T20 franchise leagues, West Indies lie eighth and ninth respectively in the world Test and one-day international rankings, while they have missed out on qualification for the 2017 Champions Trophy in England.
However, a full-strength side has reached the final of the World T20 for the second time after chasing 193 to beat hosts India in the semi-final in Mumbai.
"We knew it was going to be a massive test against a very good India team, but I said they were 10 runs short," said Sammy, who captained West Indies to World T20 success in 2010.
They reached their target with two balls to spare despite losing Chris Gayle for only five, with Lendl Simmons hitting 82 not out from 51 balls, receiving support from Johnson Charles (52) and Andre Russell (43 not out).
"Yes Gayle is our best player but we have 15 match winners," said Sammy. "Today Charles, Simmons and Russell took responsibility."
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Simmons, who only arrived in Mumbai on Tuesday as a replacement for Andre Fletcher, could have been out three times in Mumbai.
Twice he was caught off no-balls, while he was held at long-on by Ravindra Jadeja, only for the fielder to step on the rope.
"I was under pressure, the team said I had to deliver and I guess I did tonight," said Simmons.
"We didn't get the start we wanted and then I got three chances and I just cashed in."
West Indies' run to the final of the World Twenty20 is a boost to Caribbean cricket, says captain Darren Sammy.