United won with an injury-time penalty - via VAR - converted by Bruno Fernandes.
Afterwards, Solskjaer said: "We got away with one. Maybe one point we deserved, we didn't deserve more.But the character deserved more.
"Last season we had too many draws, so that is a big plus for us. You have to be happy Jose is not here to measure the goal posts.
"They are a difficult team to play against. We don't have the legs and sharpness but we will get there. We have plenty of work ahead."
Meanwhile, Brighton boss Graham Potter declared: "The performance level was amazing from us, I'm so proud of the players. It's a sore one, to say the least. Sometimes life isn't fair and it feels like that at the moment.
"All the little things have gone against us but we were really good for 90 minutes and we have to take that forwards. We have developed as a group, as a team, and we will take that setback as positively as we can.
"I don't know what the shotcount was but we were dominant. We deserved something from the game to say the least."