The Twins received left-handed reliever Tyler Watson and $500,000 in international bonus pool money in exchange for Kintzler.
The Washington Post was first to report the trade.
Kintzler, who will turn 33 on Tuesday, is in the midst of a career year, going 2-2 with a 2.78 ERA and a career-high 28 saves in 45 appearances with Minnesota, earning him his first career All-Star selection in the American League.
The eighth-year veteran right-hander is 12-13 with a 3.24 ERA and 45 saves (all with Minnesota) over 271 career games between the Milwaukee Brewers (2010-2015) and Twins.
Kintzler is set to become a free agent at the end of the season. He is owed $1 million for the rest of the season, per The Washington Post.
Watson, 20, is 6-4 with a 4.35 ERA and 98 strikeouts in 18 games (17 starts) with Single-A Hagerstown this year.