The All-Star power forward is averaging 7.3 assists a game, more than Clippers point guard Chris Paul's six assists a game, and more than the Spurs' leader in assists, Manu Ginobili, who is averaging 4.5 a game.
"C.P. draws a lot of attention, so when he comes off of it, it's my job to facilitate after that," Griffin said. "Passing is always a point of emphasis, but I think defense and rebounding are two of the biggest things I've locked in on for this series."
Griffin is also leading both teams in rebounding, averaging 13.3 a game, more than Clippers center DeAndre Jordan's 12.8, and more than the Spurs' leading rebounder in the series, Tim Duncan, who is averaging 10.8 a game.
In the Clippers' 114-105 Game 4 win over the Spurs on Sunday, Griffin finished with 20 points, 19 rebounds and seven assists.
At Monday's practice, Rivers praised Griffin's effort on the boards.
"Blake decided, well, I’m going to help the team by getting 20 rebounds," Rivers said.
The Clippers and the Spurs are tied at 2-2 in their best-of-seven series. Game 5 will be Tuesday at 7:30 p.m. at Staples Center.
"Coming into this series, it was a big thing, rebounding," Griffin said. Sunday "in particular, with D.J. being off the court for stretches there in the third and fourth, that’s just kind of what we needed. [Glen 'Big Baby' Davis] was doing a great job of keeping Tim Duncan off the boards, so somebody has to go get it. Everybody blocked out, and a lot of times I was guarding [Boris] Diaw and [Matt] Bonner, who are more around the perimeter, so I just had the opportunity."