Green's fourth three-pointer of Game 5 against the Heat on Sunday marked his 23rd three-pointer of the series, moving him one past Heat guard Ray Allen, who made 22 in six games as a member of the title-winning Celtics against the Lakers in 2008.
The record-breaking three came with 9:39 remaining in the third quarter and gave San Antonio a 66-60 lead.
"I'm keeping focus on playing basketball," Green said on Saturday, according to the Sun-Sentinel. "I'm not worried about catching any records. I'm worried about helping my team and playing well and helping us win a game. The numbers are things I don't like to reflect on or look back on. I don't want to jinx it, but at this point numbers really don't matter."
The fourth-year guard went a perfect 5-for-5 from deep in Game 2 before hitting seven three-pointers in Game 3. Green became one of only six players to hit at least seven three-pointers in an NBA Finals game.
The Spurs also set an NBA Finals record by hitting 16 three-pointers as a team in Game 3.
Green, 25, shot 42.9 percent from deep during the regular season. Entering Game 5, he was connecting on 50.5 percent of his threes during the 2013 playoffs.
The Spurs and Heat entered Game 5 at the AT&T Center on Sunday tied at two games apiece. The series will shift to Miami for Game 6 on Tuesday and, if necessary, Game 7 on Thursday. Video via YouTube user sky2847