Texas announced the deal during its game Tuesday night, just moments after Martin was seen high-fiving teammates in the bullpen before leaving down the tunnel.
Martin, 33, was 0-2 with four saves and a 3.08 ERA in 38 appearances. The right-hander has allowed only three earned runs over his past 19 games, since May 24. The 6-foot-8 Martin has 24 strikeouts and only one walk in 18⅔ innings over that span for his hometown Rangers.
"I know he might not be as well-known as the [other available relievers] with longer track records and careers, but we felt [Martin] had as much upside as anybody in the trade market, in terms of a late-inning pen arm," Braves general manager Alex Anthopoulos said on a conference call. "He's obviously gotten a chance to close some games for the Rangers. He's done [that] in Japan. His numbers speak for themselves.
"[He's a] strike thrower, very good stuff, good velocity, and someone we think can really emerge for us at the back end of the bullpen. [Manager Brian Snitker] will make the decision in terms of roles, and that's what I told Chris. He told me he has no problem [how he's used] -- he just wants to help the team. We felt like, of the options that were out there, when you factor in all of it, we think there's pretty big upside to Chris."
Martin has the best strikeout-to-walk ratio of any pitcher to throw at least 30 innings this season, according to ESPN Stats & Information research. The Braves' bullpen has struggled with walks all season, as it ranks 29th in MLB in walk rate at 11%.
Atlanta assumes the remaining $737,903 of Martin's $2.25 million salary. He has earned $100,000 in bonuses for 15 games finished, would get another $50,000 for finishing 20, and $100,000 each for finishing 25, 30, 35 and 40 games.
"I've got a couple of texts after the game about him, and everything's really positive," Snitker said. "People really like this guy, so he'll be a big addition for us."
Allard, a 21-year-old lefty, was the 14th overall pick by the Braves in the 2015 amateur draft. He was 7-5 with a 4.17 ERA in 20 starts this season for Triple-A Gwinnett. The Rangers assigned him to Triple-A Nashville.
In 2018, Allard appeared in three games with the Braves, going 1-1 with a 12.38 ERA in eight innings.
Anthopoulos said it's impossible to say whether the Braves are done dealing.
"We can improve in our rotation, we can improve offensively, we can improve in the bullpen. I'm not trying to be cryptic or vague here, but because we can improve the club in so many ways, we're not so specific in terms of a position," Anthopoulos said. "If there's a player out there that we like, that the acquisition cost makes sense, we'll pursue it."
ESPN's Eddie Matz and The Associated Press contributed to this report.