With their receiving group back to full strength this season, the Bengals (2-0) have been tough to hold down.
"It's our offense," said Dalton, who has yet to throw an interception or get sacked. "There's different things we can do. We have a good mix of everything."
San Diego (1-1) set up the first touchdown with a fumble and never led.
"He ran past guys," said Hill, who had 39 yards on 10 carries. "He ran through guys. He had a strong game."
Rivers brought a streak of 20 straight completions into the game, four shy of Donovan McNabb's NFL record. He completed his first two throws, and then was high with a third-down pass to Stevie Johnson.
He threw a 10-yard touchdown pass early in the second half, the 255th of his career that moved him ahead of Dan Fouts for the club record. Fouts was at Paul Brown Stadium as a television analyst.
"It's humbling to think about," said Rivers, who was 21 of 27 for 241 yards. "It's not as sweet right now because we lost the game. Once I've had a chance to take a deep breath and look back, I'll be thankful."
Rivers added a 40-yard touchdown to Malcom Floyd with 4:20 left, cutting it to 24-19. His final throw was intercepted by Vinny Rey at midfield with 53 seconds left to clinch it.
San Diego returned to the site of its only playoff victory of the past six years - a first-round win during the 2013 season. Bernard had a fumble in that one, and Dalton had three second-half turnovers.
The tandem made the big plays this time.
Both teams missed a field goal and fumbled twice.
Notes: Peyton Manning completed 23 straight passes for the NFL's second-longest streak. Joe Montana, Mark Brunell, David Carr and Matt Ryan had also streaks of 22 completions along with Rivers. ... Rivers' 2 TD passes moved him ahead of Sonny Jurgensen into 14th on the all-time list with 256. ... The Bengals benefited from a review late in the first half. Dalton was hit by Manti Te'o as his arm started forward, resulting in a fumble that Jerry Attaochu returned for a touchdown. After a review, it was ruled an incompletion, and the Bengals took a 14-6 lead into halftime.
Copyright 2015 by The Associated Press