Boss Nigel Pearson insisted Leicester will remain grounded despite moving off the bottom for the first time since November.
The Foxes are only in the Barclays Premier League relegation zone on goal difference after their deserved 2-0 win over Swansea.
Leonardo Ulloa and Andy King struck to clinch a third straight win 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 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 his first league strike since Boxing Day and Pearson praised the 10-goal forward after 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."
Swansea were poor and rarely looked like recovering from Ulloa's 15th-minute opener aside from a brief period after the break.
Nelson Oliveira was denied by Kasper Schmeichel and Jonjo Shelvey tested the keeper but defeat kept them eighth after Lukasz Fabianski spilled Esteban Cambiasso's free-kick for King to score in the final minute.
Boss Garry Monk felt Oliveira was too honest when he stayed on his feet after a challenge from Robert Huth before being thwarted by Schmeichel.
"The defender made a rash challenge and had he gone down it would have been a penalty and a red card which would have changed the game," he said.
"He stayed on his feet but I'm more disappointed we didn't finish the chance. It would have changed the atmosphere but credit to Leicester, they fought hard."
Swansea are aiming to set a record points total in the Premier League by beating their current total of 47 but Monk felt his men were out-battled as they struggled to match Leicester's intensity.
He said: "I felt the first half was where we really lost the game, especially in the first 20 minutes.
"We probably got outfought. We explained to the players that first 20 minutes was always going to be crucial.
"The players' effort and commitment is fantastic but we weren't quite at our best. This one defeat won't define our season."
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