In a flurry of moves Wednesday, they dealt Stewart to the Pittsburgh Pirates and Rosario to the Oakland Athletics for players to be named and sent Valencia to the Baltimore Orioles for cash considerations.
All three players had been designated for assignment on Nov. 20.
Stewart was 0-2 with a 22.24 ERA in two starts after being obtained from the Chicago White Sox on June 24 for Kevin Youkilis. The Red Sox had claimed Rosario from the Miami Marlins on Oct. 17. Valencia was 4-for-28 for Boston last season, batting .143 with one homer and four RBIs. The Red Sox obtained him from the Minnesota Twins on Aug. 5.
Mazzaro was 4-3 with a 5.73 ERA for the Royals while filling in as a spot starter because of injuries last season. Robinson struck out twice in his first four major league at-bats. Both were designated for assignment on Nov. 20.
The 21-year-old Santos pitched for both Dominican Pirates clubs last season, combining for a 6-3 record and a 2.44 ERA in 14 games, including 12 starts. In 62 2-3 innings, he allowed 42 hits, walked 20 and struck out 74 while holding opponents to a .184 average.
Rico, who turns 19 on Thursday, was 0-1 with a 7.04 ERA in 11 games for the Dominican Pirates last season.
Texas announced the deal Wednesday, eight days after the Padres designated Burns for assignment.
Burns made his major league debut, allowing 11 runs over 18 innings in 17 relief appearances for the Padres. He was 1-2 with three saves and a 3.14 ERA in 54 games for Triple-A Tucson.
The Rangers also announced that right-hander Yu Darvish was selected as the team's 2012 rookie of the year in voting by the local chapter of the Baseball Writers Association of America. Texas also agreed to terms with infielder Yangervis Solarte and right-hander Collin Balester on a minor league contracts. Those include invitations to major league spring training.