Kansas City were looking for a pass rusher after trading Dee Ford to the San Francisco 49ers and letting go of Justin Houston, and will now look for Ogbah to fill the void.
They announced the news on Monday, moments after Ogbah tweeted a celebratory emoji.
Kansas City dealt safety Eric Murray to Cleveland in the trade.
Ogbah, 25, was selected by the Browns in the second round of the 2016 draft.
He started all 16 games in his rookie season and led the team with 5.5 sacks. Ogbah had the sixth-most tackles (53) that year. His 2017 campaign was curtailed by a foot injury.
Ogbah finished last season with 40 total tackles, three sacks and eight passes defended through 14 games.
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