Leonardo Ulloa - only selected to play after an injury to David Nugent in the warm-up gave Leicester an excellent start at the King Power Stadium, latching onto a knock-down from Wes Morgan to bag his eighth league goal of the season on 15 minutes.
And the home fans' nerves were eased when midfielder Andy King made it 2-0 in the final minute of normal time.
Coupled with Burnley's 1-0 defeat at Everton, the result moved Leicester up to 18th, with only goal difference now separating them from safety.
And with a game in hand on both Burnley and QPR, plus a relatively kind run-in, Pearson's side look to have a decent chance of avoiding an immediate return to the Championship.
"The danger is that people start believing that the job is done," the manager said.
"It is far from it; there's still a long way to go.
"We have still got six games left and we have to get as many points as possible from them, which is going to be a big challenge."
Next up for Leicester is a crucial meeting at Burnley on Saturday, while their final two games of the campaign are also against fellow strugglers in the form of Sunderland and QPR.
"To get three wins on the trot in this division is very hard and doubly so considering the position we're in," Pearson said.
"We are delighted. We're now involved in a six-game season and four of those are at home.
"Our fate is still in our hands, which is nice and that's how we want to keep it for as long as possible."
