Messi had not made the trip to Atlanta, after a busy run of games since making his Miami debut in late July.
Miami coach Gerardo 'Tata' Martino said after the loss that Messi had been suffering from "muscle fatigue" and that it would have been "very reckless to bring him to play this game".
The 36-year-old sat out Argentina's World Cup qualifier against Bolivia in La Paz on Tuesday but watched his team-mates from the bench.
With no Messi last week, Miami defeated Sporting Kansas City 3-2 but this time it was a very different story for Martino's team.
Miami went ahead in the 25th minute with a superbly taken goal from in-form Ecuadoran striker Leonardo Campana.
But the response from Atlanta, sixth in the Eastern Conference, was emphatic with three goals in eight minutes.
A glancing header from French midfielder Tristan Muyumba, which struck the inside of the post, was ruled to have crossed the line by the assistant referee.
Miami got back in the game in the 54th minute through a Campana penalty but -- while they have made a habit of comebacks with Messi -- there was no such revival this time.
As they pushed forward in search of an equalizer, Atlanta caught them on the counter with Greek striker Giorgos Giakoumakis slotting home his 14th goal of the season, bringing him level as joint top scorer in the MLS.
Miami remain next to bottom of the Eastern Conference, six points behind D.C United who occupy ninth place, the last playoff spot.
Martino defended the decision to leave Messi and Alba -- who he said also had muscle fatigue -- at home ahead of a busy run of games with league matches on Wednesday and Sunday before the US Open Cup final on September 27.
"We risked worse consequences if they came to play this game," said the coach.
- Comebacks -
The visitors grabbed a point when Boupendza found Brandon Vazquez with a magnificent low ball and the USA striker found the net.
In-form Orlando City produced an even more impressive comeback at home to Columbus, winning 4-3 after trailing 3-1.
Ramiro Enrique levelled in the 86th minute with an opportunist strike and then won the game in the seventh minute of stoppage time, taking advantage of chaos in the Crew defence.
In the Western Conference, Denis Bouanga scored twice for champions Los Angeles FC as they beat cross-city rivals Los Angeles Galaxy 4-2.
Bouanga headed LAFC in front in the 23rd minute but the Galaxy struck back immediately with English forward Billy Sharp firing home his third goal in five appearances since joining the club from Sheffield United.
Bouanga's second and an 84th minute goal from Timothy Tillman clinched the derby win and allowed LAFC to narrow the deficit on leaders St. Louis City to six points.
The New England Revolution had a tumultuous week with the resignation, after an MLS investigation, of their head coach Bruce Arena and then a reshuffle of the interim coaching staff.
A win at Colorado would have secured a playoff spot but the Revolution fell to a 2-1 defeat to a team which had won just one of their last 18 games and have the worst record in the league.