Having gone 2-0 down on aggregate in the second leg at White Hart Lane on Thursday a Christian Eriksen free-kick and a quickfire Emmanuel Adebayor double turned the tie in favour of the hosts, who now face Benfica.
Next up for Tottenham is a home game against struggling Cardiff City on Sunday as they look to get back to winning ways in the Premier League following the 1-0 defeat at Norwich City last week which left them six points adrift of fourth-placed Liverpool in fifth spot.
"A club like Tottenham are up there with the top teams and I think they have to be in the Champions League," said Dembele.
"I play with these players every day and of course it is my opinion, that we need to be in the Champions League.
"There are a lot of good teams in the Premier League and we have to work hard to reach the top four.
"We want to win something, too, and if we can get the Europa League, we will take it with both hands."
There has been speculation that Tottenham boss Tim Sherwood will be replaced in the summer by the likes of Frank de Boer or Louis van Gaal, but Dembele is happy working under the 45-year-old and insists the whole squad is behind him.
"Would I like him to stay long term? That's not for me to decide," Dembele added.
"But how it looks now I think the team has confidence, we're playing much better and of course everybody is happy. After that the club has to decide, but as a team I think right now we are very happy."
Following the defeat at former club Norwich, Sherwood is keen to use the Dnipro result as a springboard for improved Premier League form.
"It is a big platform for us," he said. "On the back of a poor game at Norwich we should be bouncing into the game against Cardiff. You are only as good as your last game and on Thursday we were very good.
"It will be a tough game. Cardiff are fighting for their lives and are bottom of the league.
"They have some maverick players who on their day can win games so we will be cautious of them but buoyed at the same time by the performance against Dnipro. We should win that game."