Colchester were gifted a 20th-minute lead at Boundary Park when Andy Bond's weak shot was fumbled by home goalkeeper Dean Bouzanis, allowing Drey Wright to fire home from close range.
Both sides created half-chances in a tense contest and Oldham levelled after 68 minutes, with Jose Baxter converting a 10-yard header from Lee Croft's cross for his 14th goal of the season.
Experienced striker Jabo Ibehre went close to restoring Colchester's lead, but his header came back off the crossbar, while Baxter could have earned victory for Oldham in the 80th minute, only to volley over at full stretch.
Johnson, the Football League's youngest manager, has secured one victory and two draws since taking over at Oldham.
He said: "Their goal came from a momentary lapse in concentration, but that happens to goalkeepers.
"Nobody is being hung out to dry and it was really positive in the dressing room at half-time because everyone wanted to do it for Dean Bouzanis.
"Our atmosphere is about camaraderie and I can feel the passion of the lads in wanting to get out of trouble.
"They have got the bit between their teeth and we need to be relentless in staying unbeaten and picking up points.
"We know we aren't far away from doing something quite special, but we have to cut out mistakes and find a bit more quality.
"We go into every game wanting to win, but in the circumstances it was massive that we came back for a point."