Speaking to The Athletic, Martino recalled how captain Messi had been sorely disappointed after a Major League Soccer game with Nashville 0-0.
Martino said: "Leo was visibly frustrated after our scoreless draw against Nashville.
"I had to tell him 'relax, we're not going to win every game. We can't allow draws but we have to continue to find ways to grow.'
"He was very bitter. It was as if we had lost the game."
Messi made the move to the American side in June after leaving PSG and he has been joined by former Barcelona teammates Sergio Busquets and Jordi Alba.
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The striker already has 11 goals in as many appearances for Inter Miami but his coach spoke about the pressure Messi puts on himself to consistently perform.
Martino added: "We're still trying to find ourselves (as a team), but he's on a permanent quest to win and it's very difficult to change that mentality.
"I would never want to change that mentality either. That's what makes him better."
The 36-year-old finally completed his trophy collection after he led Argentina to victory at the Qatar World Cup, following a nail-biting final against France which finished 4-2 on penalties.
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But his next mission is to take Inter Miami to the MLS playoffs and beyond as the team have never made it past their first playoff match.
Inter Miami have eight games left to play this season, but they must make it into the top nine spots of the Western Conferences table if they are to qualify for the playoffs at all.
They currently sit 14th in the table having recorded eight wins, 14 losses and four draws this season.
But Martino remains confident that a determined Messi can help this team to great things.
He said: "I like that he's enjoying himself. But more so that he hasn't lost that will to compete.
"That's the most important aspect."