Thunderstorms and lightning delayed the start by 90 minutes, but sparks flew on the field after, with Messi, Argentina's all-time leading goalscorer and most-capped player, at the heart of the action.
Miami grabbed the lead in the 32nd minute when Brazil's Casemiro rose to head home Messi's corner.
Atlanta equalized three minutes later when former Newcastle forward Miguel Almiron blasted home a low shot.
Former Liverpool and Barcelona star Luis Suarez put Miami into the lead with a measured finish on the cusp of half-time after a move down the left started by Messi.
But Swiss striker Breel Embolo struck home an 87th-minute penalty to secure a share of the spoils.