Brighton had failed to score in their last three matches but produced an impressive display to claim their first league win away from home since February.
"It has been a tough week; especially today because we wanted a reaction [to the defeat at Crystal Palace]," Arteta told BBC. "It came in the second half, which is too late; we have to have the courage and personality to play much better and raise the level in the first half.
"We made it really difficult for ourselves. We were sloppy, we had no purpose to attack, we were really imprecise and we didn't want to play forward.
"We've shown for many weeks how well we can play; it has been a difficult week and we've lost some big players, but if you get what you want, not everything is going to be with a blue sky. This is our team in the good moments and especially the defeats.
"There are many games to play but we have to be concerned about that performance, especially first half."
Mikel Arteta rejigged the Arsenal starting lineup due to injuries and midfielder Granit Xhaka started at left-back, a decision that gave Arsenal no control in midfield and no cohesion in attack in the first half.
Brighton took the lead after 28 minutes when Trossard produced a fine finish into the top corner past Aaron Ramsdale.
The hosts thought they had equalised on the stroke of half-time but Gabriel Martinelli's goal was ruled out for offside after a lengthy VAR check.
Arsenal started the second-half brightly but found themselves 2-0 down after 67 minutes when Mwepu's stunning half-volley flew into the bottom corner.
Odegaard grabbed a goal back for Arsenal in the 89th minute but they failed to complete the comeback and suffered a disappointing loss.
"I'm glad we won today, we needed the three points. Coming from six defeats is not easy," Mwepu said.
"We really believe in ourselves, we have the right mentality in the group and we showed what we could do today. The gameplan was to win."
Information from Reuters was included in this report.