The Dodgers had faced a Sunday 5 p.m. ET deadline to sign Ryu after winning his negotiating rights with a $25.7 million posting fee.
The Dodgers are now in position to trade a starting pitcher, probably lefty Chris Capuano. The team has eight starters set to earn at least $6 million each next season, including two who are physically questionable: righty Chad Billingsley and lefty Ted Lilly.
The Rangers, who made a spirited push for Greinke, will be forced to regroup — perhaps focus on re-signing free-agent outfielder Josh Hamilton and trading for a starting pitcher.
Plan A for the Rangers was to sign Greinke and trade for Arizona Diamondbacks right fielder Justin Upton. Tampa Bay Rays righty James Shields, New York Mets righty R.A. Dickey are among the pitchers that the Rangers are considering in trades.
Beyond that, the signing of Greinke also will help establish the market for the other top free-agent starting pitchers — notably, right-handers Anibal Sanchez, Kyle Lohse and Edwin Jackson.
Greinke spent last season with the Milwaukee Brewers and Los Angeles Angels, going a combined 15-5 with a 3.48 ERA. The Angels wanted to re-sign Greinke, but balked at the price, sources said.