Matheus Cunha's first-half strike - United's only shot on target - proved decisive, condemning the Blues to a fourth straight Premier League loss without scoring, their worst such streak since March 1998.
The result leaves Chelsea sixth in the table, four points behind fifth-placed Liverpool, who also have a game in hand, while United tightened their grip on a top-five finish with just five matches remaining.
Rosenior acknowledged the scale of the task ahead, insisting his side must respond immediately.
"It's not insurmountable, but it gives us a mountain to climb," he said. "We have to go to Brighton with the idea that we must win and kick-start the rest of our season."
Frustration among supporters was clear throughout the day, with protests before kick-off and chants of "we want our Chelsea back" ringing around the stadium as the team's struggles continued.
Once criticised for defensive frailties, Chelsea are now under scrutiny for their lack of cutting edge in attack, having gone more than six hours without a league goal during their current slump.
The implications of missing out on the Champions League could be significant, both financially and in terms of squad planning. However, Rosenior admitted uncertainty over what the consequences might be.
"The honest answer is I don't know," he said. "We're still fighting and we'll address that at the end of the season."