The frontman was a second half substitute in Sunday's 2-0 victory over Leicester City in the English Premier League, following a transfer from Championship club Aston Villa last January.
“It was fantastic to get three points on my debut – I could not have asked for more,” Ayew said, as reported by his club's official website.
“My job when I came on was to make some runs to help push Leicester back. That’s what I tried to do.
“But the most important thing was that we won and moved further away from the relegation zone.
“It has been a great start for me at the club."
Ayew joined the Swans for an undisclosed fee in a swap deal that saw Neil Taylor move to the second tier side.
The Ghanaian, however, delayed his debut for the Welsh club due to his involvement in the recent Africa Cup of Nations in Gabon.
“I have had a great welcome from everybody and I am very happy to be here," Ayew added.
“The supporters have given me a great welcome too so everything is good.
“They were fantastic when I came on against Leicester. They have shown me support straightaway and now I will do my best to pay them back on the pitch.”
Also formerly of French clubs Marseille, Sochaux and Lorient, Ayew could make his first Swansea start when the Jack Army face Chelsea on February 25.
He scored twice in 21 league appearances for Villa this campaign prior to his departure.
Last season, he netted seven times in 30 league games.