When asked whether Bangladesh fancied their chances of making it to the semi-finals, Bashar said: "Yes, we do but we need to work very hard in the remaining three matches."
Bangladesh made a mockery of the rankings when they registered a 67-run victory over Graeme Smith's team in a Super Eights match here on Saturday, their first-ever win over South Africa in one-day internationals.
South Africa were dismissed for 184 chasing a 252-run target to suffer their first defeat in the second round.
"This game was important for us because we were a bit laidback," said Bashar, whose side had lost to defending champions Australia and New Zealand in the first two matches.
"It (the win) has brought a bit of our toughness back. We have a chance now. If we had lost, we would have absolutely no chance. We do have a chance now, but we have to work really hard.
"This is a big win for Bangladesh. It is more important because it shows that we are in the Super Eights not just to participate, but to show the world we are capable of pulling off big surprises."
Bangladesh had already turned the formbook upside down when they whipped India in a group match in Trinidad last month, a win that eventually knocked the former champions out of the tournament.
But Bashar said the victory over South Africa was different.
"Celebrations are not what we had after beating India. That was the beginning, but it is down a little bit this time because we are getting a little bit familiar with victories," said Bashar.
The Bangladeshi captain said his team could be a force if they kept giving consistent performances at the top level.
"We need to be consistent. We don't want to play one good game and then play three bad games. I think we need to be a bit more confident and consistent," he said.
"We have always believed we can play good cricket. We came to the Super Eights because we had done something good in the first round. But we needed to prove ourselves as well that we can play at this level."
Asked if Bangladesh had finally shed their minnows' tag after wins over India and South Africa, he said: "I leave it for the people to decide.
"Our job is to play the game in the right manner. If we can keep doing like this, keep playing good cricket, I leave it to the people what they call us."
Bangladesh will now meet England in their next match at Barbados on April 11, followed by games against Ireland and the West Indies at the same venue.
Bashar said it was a home away from home for his team which enjoyed the crowd support at the Providence Stadium.
"Everybody saw how much support they gave us in the whole game. It was like we were playing in Bangladesh. They gave us support which helped us keep going," he said.
"People have supported us since we have arrived here. Wherever we go, people have told us they are a little bit upset because India is not there, but they love the way we play."