Hart produced a stunning one-handed save to deny Federico Fernandez from equalising late on, before making another point-blank stop from Bafetimbi Gomis as City claimed all three points at the Liberty Stadium.
“It was a beautiful save in a very important moment,” Pellegrini said after his side’s fifth straight league win.
“Joe is the No 1 goalkeeper in England and I think he has improved a lot this season.
“Bringing Willy Caballero was good for him, giving him good competition and also working with the goalkeeping coach Xabier Mancisidor.
“It’s important for him but the most important thing is he has an open mind. He is only 28 years old and he must improve in a lot of things, but he is working hard to do it.
“It is difficult for a young goalkeeper to be so good at the start of his career and it is difficult to compare him with all the other goalkeepers here.
“There are three or four good goalkeepers in this league, but Joe must improve and not think he knows everything at this moment. Every day he works hard.”
James Milner scored City’s second goal and gave another performance full of endeavour and commitment, even though he is out of contract at the end of the season and he may move on this summer.
“It’s a decision James must take at the end of season, what is better for his future,” Pellegrini said.
“I always talk that he is an important player for our team, he can play in different positions because he is an intelligent player.
“He always gives 100 per cent but it will be sorted at the end of the season.
“I can think (staying at Manchester City) is best him for him but it’s important what he thinks.”
Asked if he wanted to keep Milner, Pellegrini replied: “Yes, of course. He is a very important player for us.”