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The Magpies remain in the drop zone, one point from safety, after losing the Spaniard's first game as their manager.
United have nine games remaining, the next of which is at home to relegation-threatened rivals Sunderland.
Benitez, who replaced the sacked Steve McClaren on Friday, said: "With this commitment on the pitch we can do it."
Newcastle began brightly against the league leaders, but fell behind to a Shinji Okazaki overhead kick and were unable to equalise despite both Ayoze Perez and Moussa Sissoko going close.
The former Liverpool and Real Madrid boss added: "We started really well, but the reaction of the team in the second half was pleasing because you could see the commitment of the players and we were trying to win.
"When you are in this position, it's important to create and I think we'll take our chances next time."
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Newcastle's next two games, against fellow strugglers Sunderland and Norwich, are central to their survival hopes.
Sam Allardyce's Sunderland, who visit St James' Park on Sunday, are a point better off but clear of the relegation zone on goal difference only.
Benitez said: "We have to improve in both areas and that is our job now.
"We have one full week and we need to improve these little things. It's a question of having a bit of luck, but we have to think about the positives and we are creating chances.
"We know our next game against Sunderland will be important for everyone, but I have a feeling the fans are really good and they are the number 12 for the next game."
Leicester manager Claudio Ranieri, whose team are five points clear at the top of the league, supported Benitez's optimism.
"Their players showed Rafa their attitude and the Newcastle players are very good," said the Italian. "I am sure that they will be safe."
There were enough positives in Monday's 1-0 defeat by Leicester to suggest Newcastle can avoid relegation from the Premier League, says Rafael Benitez.