The former Watford man arrived in south Wales for a club record fee of £3.5million in the summer and there were high hopes that he could be the man to fire the Swans to Premier League survival having found the net 27 times last term.
And with Swansea's lack of goals during the early weeks of the season becoming a talking point the spotlight was, rather unfairly, turned on the Middlesbrough product.
But the 26-year-old retained his belief in his own abilities and marked his return to the starting line-up after a back complaint by getting the better of Jonathan Woodgate before lifting the ball over the advancing Asmir Begovic.
Graham left the field to a standing ovation moments later, but was keen to deflect any personal glory onto his team-mates.
"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."
Graham heaped praise on a team display which saw Swansea, so often acclaimed for their attractive passing style, show they have plenty of substance as they dealt with an aerial bombardment when Stoke pushed for an equaliser after Scott Sinclair had given the hosts the lead from the spot.
"When people look at the game they will see things like the amount of crosses we won in the air," he said. "We are not a massive team, most of our guys are under six foot and to match that physical challenge and defend ever so well shows a different side to us.
"We are nice on the eye with our passing and movement so it is a great result and everyone is buzzing and hopefully it sets us up for the trip to Norwich."
While the win over the Potters maintained Swansea's unbeaten home record, as well as a defensive record that has yet to see a visiting side score at the Liberty Stadium, they have yet to pick up any points on the road.
With their first three away fixtures being at Manchester City, Arsenal and Chelsea that is understandable, but with trips to Norwich and Wolves coming up after the international break Graham hopes the Swans can get positive results.
"If you look at our away games we have had Manchester City, Arsenal and Chelsea which are very difficult games," he said. "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."
Stoke manager Tony Pulis was disappointed by his side's slow start and while he did not use his side's Europa League exertions in midweek as an excuse, he admitted his squad need to find a way to adapt to the demands of the competition as this was their second straight league defeat following a European fixture.
He said: "We have had two slow starts this season, and they have been against Sunderland and Swansea, so we have to look at it and see what it's about. The frustrating thing is every European game on a Thursday, the next game in the league is away.
"I am not saying the FA or Premier League have done anything suspicious but it does not help. It's our first taste of European football and we want to embrace it as everyone, the football club and the supporters, wants to relish the chance of being in a big European competition."