The Caribbean side bowled the co-hosts out for a paltry 58 in just 18.5 overs -- their lowest ever score in a one-day international -- before cantering to their target in the 13th over of their innings.
Sammy praised the experienced Gayle for his "motivational speech before the game."
"It was a brilliant performance by my team-mates. Chris Gayle wasn't feeling well today but I thought he came out and showed how committed he is to the cause," he said.
"He said 'do not take Bangladesh lightly'. We were a little flat in the warm-up and he didn't like that so he encouraged us to go out there and remember Bangladesh would be coming hard at us."
The skipper said he was delighted at the start his side had, dismissing the dangerous Tamim Iqbal in the first over and he was happy personally to take three wickets.
"We went out there. We knew how crucial this game was for us," he said.
"We've always fancied ourselves but it's up to us to go out there and play competitive and consistent cricket."
Gayle, who scored 37 not out as he passed 8,000 ODI runs, said: "I was expecting a bigger occasion than this but having said that the wicket seems to be doing a bit."
"Kemar Roach (3-19) did a brilliant job and all the bowlers stepped up to the plate."
Disappointed Bangladesh skipper Shakib Al Hasan was at a loss to explain his team's disastrous showing, adding that he understood the fans' frustrations.
"We have a lot of things to work on now. We have to sit down and think about it but hopefully we will come back strongly."
But he said reaching the quarter-finals was still possible after an earlier defeat against India and a win against Ireland.
"It's very much possible but we have to come back harder now and do the right things," he added.
He said the next game against England in Chittagong was a must-win.
"It's a must-win game for us. We know in Chittagong it's a different sort of wicket."
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