Boss Nigel Pearson insisted that Leicester would keep their feet on the ground despite moving off the bottom of the Premier League for the first time since November.
The Foxes are only in the relegation zone on goal difference after a deserved 2-0 win over Swansea.
Leonardo Ulloa and Andy King struck to clinch a third straight victory for the Foxes -- the first time they have won three in a row in the top flight for 15 years.
A three-goal victory would have seen them escape the drop zone, but they climbed off the foot of the table and are level on points with 17th-placed Hull.
Pearson said: "We've given ourselves a more realistic chance. We have six games left, four at home, and we have to perform well in those games to retain our status.
"The danger is people get carried away. We are still in a difficult situation and we have to make sure our fate remains in our own hands as long as possible.
"The Premier League is a really big ask, mentally and physically, and we have performed well for the most part -- we've had to in order to win three games on the trot. Ultimately, the performance warranted the result.''
Ulloa, who was initially named on the bench, was a late replacement for David Nugent after the former England striker injured his calf in the warm-up.
It was 10-goal forward's first league strike since Boxing Day, and Pearson praised him for his swift impact.
He added: "The situation we're in demands everyone is switched on and tuned in to what we're trying to do. The players have to put to one side their own personal world for the good of the group.
"Leo has had a really good season for us. I'd like him to score goals but as long as he achieves what we set out to achieve I'm not too bothered.
"We can't afford for players to pace themselves -- we need them playing flat out."