The Africa Cup of Nations champions - without injured star forward Sadio Mane - needed to win to make it through from Group A, while a draw would have been good enough for Ecuador.
After seeing Watford forward Ismaila Sarr put Senegal in front from a penalty at the end of the first half, Brighton midfielder Moises Caicedo pulled the South Americans level.
However, Chelsea defender Kalidou Koulibaly soon put Senegal back in front with the captain's first international goal.
A possible showdown with England awaits in the last 16, but whichever side Senegal face Cisse accepts his team will need to quickly refocus for the business end of the tournament.
"We don't know who we're going to come up against in the round of 16 - it could be England, it could be Wales, Iran, USA, anyone," Cisse said.
"Then that will be a new stage for us. We know that we are moving to the knockout stages, which is completely different from what we are used to playing in these types of (group) matches.
"Everyone is a good team, the best teams in the world that are competing. Therefore, we need to be prepared to come up against any side.
"It is now a win or lose situation, there are no second chances - if you win, you go through, if you lose, you go home.
"So for the moment, we are just going to concentrate on resting and getting fit for the next game, it is simple as that."
After securing qualification, Senegal's players paid tribute to Papa Bouba Diop on the anniversary of his death two years ago.
The squad held up a flag of the former midfielder, who scored the country's first goal at the 2002 World Cup where they went on to reach the quarter-finals.
"We knew this game was the most important or one of the most important for our career," Chelsea defender Koulibaly said.
"This (player of the match) trophy, I will give it to the family of Papa Bouba Diop, because this was a special day for us, the anniversary of his death."
Koulibaly added: "We want to show exactly why we are the champions of Africa and why Senegal should be playing on the world scene.
"We have to respect the next team we will play against and we will wait to see which team we can play against.
"But the team who will play against us can be afraid a little bit because they see that Senegal is a good team with a lot of talents and we can do good things."
After picking up four points from their opening two games, Ecuador coach Gustavo Alfaro accepted his side had come up short when it mattered.
"It is a hard blow and big frustration. We knew that margin for mistakes was really high, that details could make all the difference and this is what happened," Alfaro said.
"In the first half, we didn't have enough chances and missed the second balls. During half-time, I told them if you want to equalise, you have to do something different and need to play.
"We did that more in the second half and drew level, but after having worked so hard, we were then not able to control the pace of the game.
"We wanted to go through on this road that we have been following and the pain we feel, like the whole of Ecuador, is huge pain.
"But we must not forget that this was a good performance from this generation of players and the experience will be useful for the future."