The Argentine told his team that they had an "opportunity" at the interval despite Conor Gallagher getting sent off and Facundo Buonanotte pulling a goal back for the visitors at Stamford Bridge.
Enzo Fernandez and Levi Colwill had each scored headers from corners to put Chelsea into a comfortable position. Pochettino explained how he lifted his vulnerable-looking team after 45 minutes.
"At half-time, I told them it is 2-1 and these things happen," Pochettino said. "It is a great opportunity and a massive challenge for us to show we are a team and it will improve us.
"I feel proud because, after [losing at] Newcastle, a team like Brighton is difficult to play; the first 20 minutes, we scored twice and controlled the game.
"We concede and Gallagher was sent off but until the end, we didn't concede too much and scored another goal.
"We are Chelsea and Chelsea expect to play for a win, so the period we are in now, it is not easy to play.
"That's why we are building momentum step-by-step and I am proud of the character we showed."
Pochettino was critical of referee Craig Pawson after he sent off Gallagher, saying: "I don’t think it is a sending off, but I may be wrong.”
Fernandez scored a crucial penalty in the second half against Brighton, and Pochettino added: "Why was the penalty of [Mykhailo] Mudryk waiting for VAR and he gave the [Colwill] handball? It is so weird the way we use VAR sometimes."
Pochettino criticised the decision to award a penalty to Brighton in the last minute after Joao Pedro had pulled another goal back.
Pawson thought the ball had struck Colwill's arm, but VAR corrected the decision after showing it had clearly hit his head.
"Why did they give the penalty and then check VAR [for the Colwill handball]? It is difficult to understand why," he added in a separate interview with BBC's Match of the Day.