Fletcher produced goals either side of the break as Sunderland left Swansea City with their second Premier League point of the season after drawing 2-2.
The 25-year-old told BBC Sport: "I was delighted to get off the mark. It takes a lot of pressure off to get the two goals.
"The manager told me there's no pressure on me and hopefully I've repaid him with the goals today."
Martin O'Neill signed Fletcher last week for £14m from Wolverhampton Wanderers.