With one of the deepest backfields in the NFL, the Broncos gained 205 of their 303 yards on the ground. Anderson led the assault while Olandis Gary added 61 yards and a touchdown.
"It was a three-headed monster," Griese said. "They held onto the ball after taking some big hits. The offensive line made some openings. We were a team today. We want to dominate people and win by big numbers.
"The 49ers are a really good defense. To come out and establish a running game was big for us."
It was the second straight impressive win for the Broncos (2-0) against a league power after they posted a 23-16 victory over NFC champion St. Louis in their opener.
"I know it's an old coaching cliche, but you have to stay focused week-to-week," Broncos Coach Mike Shanahan said. "It's a long season. If you don't do it, that's the way teams sneak up on you."
Griese, who nearly was pulled in the opener, completed 14 of 19 passes for 119 yards. He had touchdown passes of one yard to Anderson and 11 yards to Scottie Montgomery.
Jeff Garcia passed for a touchdown and ran for a score for the 49ers (1-1), who committed 10 penalties for 114 yards and were hurt by two turnovers.
"We had far too many penalties today," 49ers Coach Steve Mariucci said. "Penalties were lopsided. They didn't have very many and we had a lot. We can't make that as an excuse. Good teams make their own breaks."
Trailing, 7-3, the Broncos made the necessary adjustments at the half and moved 74 yards in 10 plays on the opening possession of the third quarter to take the lead for good.
Shanahan called on the ground game as the first six plays were runs that moved Denver into San Francisco territory.
After an incompletion on first down from the San Francisco 45, Anderson ripped off a 15-yard run up the middle. Rookie Clinton Portis gained a yard before Griese found Rod Smith for 12 yards to the one.
The Broncos' top receiver, Smith had just three catches for 22 yards. Griese looked often for tight end Shannon Sharpe, who had five catches for 52 yards.
Griese used a play-action pass and found Anderson for the touchdown,
giving the Broncos a 10-7 lead with 9:06 left in the quarter.
Even though Garcia completed 27 of 36 passes for 205 yards, his club's offense was out of rhythm for much of the afternoon against the Broncos' attacking defense.
San Francisco failed to move on its final two possessions of third and Gary capped a four-play, 46-yard drive 40 seconds into the fourth with a one-yard run to give Denver a 17-7 lead.
Garcia moved the 49ers into Broncos territory on their next possession, but on first and 10 from the 36, he was intercepted by cornerback Deltha O'Neal at the one with 9:36 left.
In addition to containing Garcia, the Broncos also shut down the 49ers' running game. Kevan Barlow carried seven times for 47 yards while Garrison Hearst was limited to five yards on five attempts.
"At times we showed we could run the football, we just didn't do it enough," Garcia said. "We needed to execute. You can't afford to be second and long, third and long situations. If we don't make the mistakes like we did today, it's a more competitive ballgame. We made too many mistakes."
"What killed us was our inability to sustain drives the way we should," Mariucci said. "We've been a little off here and there on our timing. But we need to start hitting on all cylinders, especially against these good teams. The team knows we have a very challenging schedule with a lot of good teams back-to-back. We have to bring our 'A' game."