The seven-times Ballon d'Or winner then doubled the advantage in the 22nd minute, the first of three goals that saw him combine with Finnish winger Robert Taylor.
Messi ran at the back-pedalling Atlanta defence before switching to Taylor who slid the ball back into the path of the 36-year-old who slotted home.
Another Busquets ball over the top found Messi but the off-side flag was raised after he shot straight at Atlanta keeper Brad Guzan.
Miami made it 3-0, a minute before the interval when Messi fed Benjamin Cremaschi, who cleverly flicked the ball to Taylor who defied the tight angle with a fierce blast.
Eight minutes after half-time, Atlanta were caught on the break when a corner was cleared to Messi, who ran from deep in his own half before feeding Taylor to his left who drilled in a low shot off the inside of the far post.
The margin of victory allowed Miami's Argentine head coach Gerardo Martino to rest Busquets and Messi with the latter substituted in the 78th minute.
Messi's Argentina team-mate Thiago Almada had a chance for a consolation for Atlanta but his penalty was well saved by Miami keeper Drew Callender.
Miami, bottom of the MLS standings, had not won for 11 league games prior to Messi's arrival, but looks utterly transformed by his impact, together with Busquets.
"Their arrival has given the players a lot of confidence," said Martino, who ten years ago coached the pair at Barcelona.
"The players feel more supported, they feel that the responsibility is carried by the great players and so they feel more liberated to play," he added.
Martino, who also coached Messi with the Argentine national team, said his World Cup triumph last season had given him a new lease on life.
"He has taken a huge weight off his shoulders with the Argentine national team. You can see it in everything, in his day-to-day life, in his statements, in the way he approaches the games. Leo is liberated and we can expect more of this type of performance," he added.
Atlanta were MLS champions in 2018 under Martino but their current boss, Gonzalo Pineda said they had been outclassed.
"They were the better team...at key moments they did not forgive. Against players like Messi, you can't make mistakes and we made them in the first, second and fourth goals," he said.
Taylor, who exploited perfectly the space created for him by Messi, deservedly received his own ovation as he left the field.
"Every time Leo has the ball we have a few players running in behind and that creates loads of space for everyone else," Taylor said.
"I mean, he's the best player in the world. This is what he does – he scores two goals, he scored last game, gets another assist. It's a dream come true to play with him," he added.
