The Boston Red Sox on Tuesday acquired middle reliever Anthony Varvaro from the Atlanta Braves in exchange for minor-league reliever Aaron Kurcz and cash considerations, which would appear to close the door on free agent Burke Badenhop’s return to the Sox. The Braves had designated Varvaro for assignment on Monday.

Varvaro, 30, made 61 appearances for the Braves last season, most as a bridge to the team’s late-inning relievers, and thrived in the role for the second consecutive season. In addition to posting a 2.63 ERA, Varvaro struck out 50 batters while walking just 13 in 54 2/3 innings. He held opposing hitters to a .228 average and was especially effective against left-handed hitters, holding them to a .149 average (11 for 74), with just five extra-base hits.

Varvaro, who grew up on Staten Island and was drafted out of St. John’s University, relies primarily on his fastball, which sits at a tick above 92 miles an hour, and a pretty good curveball, mixing in a changeup on occasion. His groundball-to-fastball ratio was virtually a 50-50 split.

The Braves DFA’d Varvaro to open a spot on their 40-man roster. He was paid $515,000 last season and is not yet eligible for salary arbitration, but in Kurcz the Braves are receiving a 24-year-old pitcher who profiles in much the same way.

Kurcz was used as a reliever in Double-A Portland last season after missing all of 2013 because of Tommy John surgery. He had come to the Red Sox from the Cubs as part of the Theo Epstein compensation package and was impressive in his first season in Double-A, striking out 72 batters in 50 1/3 innings before requiring surgery.

Last season, Kurcz was 3-2 with a 2.14 ERA for the Sea Dogs. Because he was not on Boston’s 40-man roster, he was eligible for the Rule 5 draft but was not selected.

Badenhop was outstanding for the Red Sox in 2014, posting a 2.29 ERA in 70 appearances, but was paid $2.15 million last season and is in line for a raise as a free agent.