The Englishman, signed after he was released by Watford this season, has scored three goals in his last six games.
Potter said, "The key thing with Danny is that he wanted to come, that he wanted to play, that he wanted to be part of what we are doing here, he wanted to influence the team on and off the pitch.
"He brings a fantastic quality as a footballer and as a guy.
"He is someone that the players look up to, they like, they respect.
"When you have that combination for us it ended up being an easy decision.
"We are delighted with that and credit to the club and credit to the guys working behind the scenes because they made it happen and we have all benefited from having Danny Welbeck around."
Potter also said that the 'intention' was there to retain Welbeck beyond this season.
He added: "At the moment I'm sure if you ask Danny he will want to finish off the next four matches, he will want to play the next four matches and he will want to add to his goal tally and he will want to add to the points tally of the team. That's what we will have to focus on."