Anthony Knockaert's stoppage-time goal sealed three points for the Foxes and the news from Bolton, who could only draw 2-2 with Blackpool, meant that they moved into sixth on goal difference.
That means an extension to Leicester's season as they face Watford over two legs for a place in the Wembley final later this month.
Pearson said: "The players have to take a lot of credit for dealing with the situation.
"We couldn't have got off to a worse start, but we responded very well and went on to create the best chances of the first half and were deservedly ahead, in my opinion, going into the break.
"It was like a cup final today. There was a lot of pressure on the players - for both sides - and I'm really pleased with the way we coped.
"Some of our results coming into this weren't the best, despite feeling that our performances deserved more, but we worked hard to get another opportunity so well done to the players.
"Today we had an outside chance and we've done what we needed to do in terms of winning our game. That's all we could have done and thankfully we got the help we needed from elsewhere.
"Watford have beaten us twice this season but I don't think that will have any bearing. They will be disappointed at missing out on automatic promotion but I know (Hornets boss) Gianfranco Zola and he won't let that bother them.
"The important thing is, although there's always interest in routes into play-offs and the psychological effects when dealing with disappointments, etc, keeping your nerve in those games.
"The side that holds their nerves and has a bit of luck along the way will ultimately prevail."