Sunderland have booked a place in the Capital One Cup final and Saturday's 1-0 win over Southampton sent them into the quarter-finals of the FA Cup.
This success could be overshadowed if the Black Cats slip out of the top flight and they currently sit third from bottom in the table.
Sunderland's progress in the cup competitions could complicate their fixture schedule, but O'Shea believes his team-mates will focus on the crucial upcoming games in the league.
"Without a doubt," said the defender. "That's going to be key because you never know the way the games fall and with the cup run, we could have a couple of games in the league and different situations, so it's a time for cool heads.
"From the manager's point of view, he has dealt with that extremely well so far and it will be the same going forward, I'm sure.
"But the competition amongst the squad is getting fantastic too and the manager is going to be facing some tough choices, which is always good because competition improves everyone all the time and performances have to improve alongside it.
"Winning breeds confidence and we have to take that forward. But first and foremost, we are back in training on Monday and we get back down to the business of Arsenal.
"There's lots to look forward to, they are big important games. But you have to enjoy these moments as a player too."