The Swans failed to hit the back of the net in their opening four Premier League matches but have returned to goalscoring form with six in their past three games which has helped them to two victories over West Bromwich Albion and Stoke City.
Record signing Graham, who joined from Watford in the summer, grabbed his first for his new club on the weekend but he says he never felt the pressure of beginning with a barren run.
"I am not going to lie, I didn't let it (the lack of a goal) get to me that much," he told Press Association Sport.
"It was only five Premier League games that I had played in and I have been longer without a goal at points in my career.
"I have always said if I get into the position and get the chance I am going to stick it in, I am confident in that. This was the day but the most important thing was the three points."
And Graham is sure that recording two wins in their past three outings will lift the confidence when Swansea travel to fellow top flight newcomers Norwich City on October 15.
"If you look at our away games we have had Manchester City, Arsenal and Chelsea which are very difficult games," he added. "But we go to Norwich on the back of a 2-0 win and it's important we start to pick up those points on the road as we are going to need them sooner rather than later."