The 25-year-old former Watford striker signed for the Swans for a £3.5m fee and insisted his heart was set on a move to the Liberty Stadium as soon as he heard Rodgers' plans for the club.
"When I spoke to Brendan he came across as a really enthusiastic manager," Graham told talkSPORT. "[He is] driven and wants to do well similar to myself as a player. I want to achieve goals and do well in the Premier League.
"Obviously the style of play was massive. I've been playing against them for a few years now and it's something that attracted me to Swansea.
"I met West Brom, I met QPR - it could have been quite easy for me to go to QPR I could have stayed where I was living. I spoke to West Brom and liked what I heard but not as much as Swansea. From the moment I met Brendan to leaving the meeting I knew my heart was set on Swansea City."