The 31-year-old striker netted the opener in Real's 2-0 victory at Numancia in the Primera Liga on Saturday, pouncing to put away a loose ball after the goalkeeper could only parry a Gonzalo Higuain shot in the 48th minute.
"The important thing was that we secured the victory, but on a personal level I am very pleased," club captain Raul said after the game. "I am proud to have drawn level with Di Stefano, now I hope I can overhaul him.
"I am in debt to all the coaches I have had, and all the colleagues who have helped me to make it to where I am."
Raul notched his first goal for the nine-times European champions against city rivals Atletico Madrid in November 1994 when aged just 17.
His tally includes 214 league goals, 18 in the King's Cup, and he is the Champions League's leading scorer with 64.
"It's nice that everyone congratulates me," Raul added. "But I am focused on the next match and another victory. I want to win more titles."
Raul has won three European Cups and six league championships and is in his 15th season with Real Madrid.
The Argentine-born Di Stefano helped steer Real Madrid to five consecutive European Cups between 1956 and 1960, and won eight league titles with the side. He was made the club's honorary president in 2000.
He told Antena 3 television: "I am sure Raul could score 400 or 450, he is still young. He's 31. He could play until he's 37 or 38."
'Obviously Messi is the best player in history and being compared to him means that you're doing things right, but I try to be myself': Barcelona star responds to comparisons with Argentine legend
'Darwin Nunez would be gold for Arsenal': Liverpool legend claims Gunners should have signed Uruguayan striker in January
