Baltimore's Justin Tucker kicked a 43-yard field goal to put the Ravens up by 10.
The Browns' last chance ended when safety Ed Reed intercepted Weeden, who went 20 of 37 for 176 yards and two interceptions.
"We had a chance." Weeden said. "We just couldn't finish it."
Flacco finished 15 of 24 for 153 yards. He completed his first 10 passes, but Cleveland's defense shut him and the Ravens down for most of the second half until the final TD drive.
"We started fast and finished strong," Harbaugh said. "In the middle? Eh, it was iffy. But we came together at the end to win. "I was thinking (when the Ravens were struggling, 'This is not good. It would be nice to get a first down. They didn't do anything different. They just did it better. The play to Anquan got us going."
Rice had 98 yards and an 8-yard TD run and Bernard Pierce scored from 12 yards as the Ravens jumped to a quick 14-0 lead.
Richardson is the first rookie to run for more than 100 yards against the Ravens since Jacksonville's Fred Taylor in 1998. Following the game, he and Rice exchanged jerseys on the field.
"It felt good to trade jerseys with a beast in Trent Richardson," Rice said. "Mutual respect on both ends."
Down by two TDs, the Browns' offense awakened in the second quarter, when Dawson's three field goals pulled Cleveland within five at halftime.
Dawson's 29-yarder with 3 seconds left made it 14-9 and sent Cleveland's players to the locker room feeling confident after nearly letting the Ravens get too far in front.
But it was Cleveland's inability to score TDs when it had the chance that doomed the young Browns and coach Pat Shurmur, who will head into their bye week stinging from another game they could have won.
"You start getting threes instead of sevens when you make your way down there, the outcome is much, much different," said Shurmur, who fell to 6-19 in two seasons. "When we did score, we were lined up improperly. There was some sloppiness in there. Of course, I'll take responsibility for that."
NOTES: Dawson made kicks of 32, 28, 29, 33 and 41 yards. His 23 consecutive field goals are still 19 shy of the NFL record set by Mike Vanderjagt, who converted 42 straight for the Colts. ... Weeden's interception in the third quarter snapped a streak of 209 consecutive offensive plays without a turnover for the Browns. ... Ravens CB Cary Williams picked off Weeden and has two of his four interceptions this season against the Browns. The five-year veteran entered the season with no picks in 39 career games. ... Browns executive vice president Bryan Wiedmeier attended his first game since undergoing surgery on Oct. 26 to have a brain tumor removed.
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