A sixth-minute goal by defender Joe Walsh proved enough to send the Latics down to their fourth defeat in five games.
Johnson said: "We had four or five proper chances but we've come away without a point at least. Again, some soul-searching needs to be done.
"We just have to keep going, that's all we can do. The lads must do it themselves, sometimes you need to find that character from within.
"It was a typical League One game, quite physical, and we are not doing enough to get the ball over the line. Charlie Macdonald could have got a hat trick."
Oldham have conceded the first goal eight times in nine league games and Johnson is seeking a response for the home clash against league leaders Leyton Orient.
He added: "Facing Leyton Orient will be tough but I've asked the players for a reaction. They must be man enough.
"In the second half at Crawley our final ball lacked a bit. We created four good chances but we didn't score."