The Black Cats have struggled this season and headed to St Mary's in 16th place, just a point and a place better off than the hosts. Fletcher's first half strike proved the difference in appalling conditions on the south coast as Martin O'Neill's side eked out a victory to make it two wins from their last three matches.
The Scotland international said: "We know we've got a hard Christmas period coming up but we've built a bit of confidence and we'll take that into the next game."
He added: "The boys are all very tight. We all get on very well. It is a good bunch of lads. If we keep our heads down and keep working hard, we'll get results like this."
Fletcher's instinctive strike was the summer signing's eighth league goal of the campaign.
The 25-year-old has shone for Sunderland and would like nothing more than to add to his tally when City visit the Stadium of Light on Boxing Day.
"I think Sessy [Stephane Sessegnon] will take an assist for that one but I think he had a shot," he said with a smile.
"I am just glad I got another goal and we got the win.
"It just came to me quick and my instinct was to side-foot it away. Luckily for me it went in the back of the net.
"It would be nice to get a goal under my belt and another win, but we'll wait to see what happens."