Wilson gave up an eighth-inning grand slam to Richie Sexson in the game.
Wilson was in a non-save situation to work out some mechanical problems with his delivery. After the game, Wilson admitted the elbow has been bothering him, and he'll fly to Los Angeles on Wednesday to be checked out by Dr. Lewis Yocum.
"I tried to pitch through it for a while," Wilson said. "I can't do the team much good unless I get healthy. We've been trying to treat it for the last couple of days and it hasn't worked."
Eddie Guardado got the final three outs on Tuesday night for his third save in as many chances and 186th of his career. Manager Ron Washington said Guardado will be the primary closer in Wilson's absence.
In five of his past 11 appearances, Wilson has allowed more than one run. He's given up 13 earned runs in his past 11 outings, increasing his ERA from 4.34 to 6.02.
"We had a chance to use him in a less-pressure situation but he wasn't effective," Washington said of Tuesday night's outing by Wilson. "This is the best thing for his peace of mind."
The Rangers will call up right-hander Joaquin Benoit from Triple-A Oklahoma to take Wilson's roster spot.