City have failed to shine in their opening two games, with Saturday's 3-0 loss to Middlesbrough posing questions of how equipped they are to cope with life after Roberto Martinez.
But Britton is convinced the Swans, now managed by Paulo Sousa, can quickly get back on track, starting with Tuesday's visit of Reading.
"We've got to make sure we all keep our heads up and we all go again on Tuesday against Reading," he told BBC Sport Wales.
"Obviously we're low at the moment but we'll pick ourselves up. We've got to wipe our memory clean and go into Tuesday in high spirits and confident we can get a win."