The 26-year-old stopped Connelly in round five in his hometown of Hull on Saturday night.
After the fight, Olympic champion Campbell was full of praise for his opponent.
"That's exactly what I needed," Campbell told Sky Sports News. "All credit to Lee, what a tough guy he is, he's a little warrior and I just couldn't break him. He'd have gone to the end, I'm sure he would.
"It was a great fight as I needed some rounds. But it's only my third fight and I'm learning all the time.
"I think the referee definitely made the right decision because he was taking a fair few shots."
Campbell has won all three of his professional fights to date.