The Hull forward handed the Tigers the lead with a fine far-post finish in just the eighth minute on his first return to the Liberty Stadium.
Before the match, the on-loan Sunderland man had made 13 appearances without finding the net this season, but his lean patch went back much further than that, numbering 30 games for three different clubs since he struck for Swansea in a 2-0 League Cup victory over Chelsea.
Danny Graham: "It's been a long time coming, the goal, we had to work hard and we knew it was going to be tough. They kick the ball very well.
"We set out with a shape which we thought would frustrate them on the counter-attack and that seemed to work.
"We're very happy to go home with a point.
"I wanted to go crazy. I have a great relationship and had some great times here in south Wales so that was always the case that if I scored I wasn't going to celebrate."