Suarez and Alba give Miami a winning start in MLS Cup playoffs
Miami could book their place in the Eastern Conference semi-finals if they can win game two of the best-of-three series in Atlanta on November 2 but after a tight game they will be wary of the task that awaits in Georgia.
Inter, the top seed and favourites in the playoffs, made an electric start to the game with Luis Suarez firing them ahead in the second minute when his former Barcelona team-mate Alba found Diego Gomez and the Paraguayan slipped the ball to Suarez who made no mistake from close range.
Atlanta qualified through a wildcard game in Montreal on Tuesday and they were slow out of the blocks as Miami dominated possession and gave visiting keeper Brad Guzan a busy opening half.
Alba fed Messi, whose sweetly struck, first-time shot forced the 40-year-old keeper into a fine, diving save.
Atlanta suffered a blow when right winger Brooks Lennon had to go off in the 21st minute and five minutes later Messi threatened again with his effort from 25-yards forcing another save from the outstanding Guzan.
A minute later, Miami should have doubled their lead from a corner but David Martinez headed against the post with the goal at his mercy.
Messi then hit the post with an attempt to beat Guzan at his near post and a thundering follow-up shot from Marcelo Weigandt had the former Aston Villa keeper at full stretch.
Atlanta's problems continued with defender Stian Gregersen forced off with a muscle injury but just when it seemed everything was conspiring against them they got back on level terms in the 39th minute.
Pedro Amador played a clever little pass to find Georgia winger Saba Lobzhanidze inside the box and his first-time, left-foot shot flashed into the far bottom corner.
The visitors then had a glorious chance to go in with a lead at the break when Russian Alexey Miranchuk carved out a lovely ball through to Jamal Thiare but his shot lacked composure and was well wide.
The unique format of the first round of MLS's playoff, which go straight to penalties if the games are level after 90 minutes, would have given Atlanta hope that they could pull off an upset.
But their determined efforts were undone on the hour when they fell asleep at a Miami corner; after a short corner, Messi played the ball across to Alba, who was left in acres of space outside the box.
The Spanish full-back gratefully accepted the invitation and unleashed a fierce drive from 25 yards out which flew past the helpless Guzan.
Messi had a chance to get on the scoresheet in stoppage time but he was denied by Guzan, rolling back the years, who raced out quickly to smother the shot.
Miami coach Gerardo Martino said Alba's goal was a reminder of his class as an attacking full-back.
"It's not easy to find full-backs who are so involved in attacking play. You'd have to go back to Dani Alves, Cafu, Marcelo, Roberto Carlos.... that's the level," he said.
"He has a technical quality that is very rare to see in a full-back and this is enhanced by the understanding he has with Luis and Leo. It's difficult to find a player in his position who is so decisive," added the former Barcelona coach.
Martino also had a word for former US international Guzan.
"If it wasn't for Brad's performance, we would have ended up with a few more," said the Argentine coach, who won the MLS title during his time with Atlanta.