Argentine striker Piatti scored on a cross from Andres Romero in the 16th for the Impact.
"I think it's a great result for us," said Montreal midfielder Nigel Reo-Coker. "It was not an easy place to come and get a result. At times, we played some great football and we showed what we were capable as a team. We could have done a little bit more some other times, but in the end, we kept ourselves in it."
Oribe Peralta, a second-half substitute, tied the game in the 89th for America on a header off a cross from Argentine midfielder Rubens Sambueza.
"We were compact tonight and we knew they would have more possession," added Impact forward Dominic Oduro. "The main focus was to hit them on the counter and we did that well. We showed a lot of composure as a team and we're going back to Montreal with hopes of winning it at home."
Montreal needs a scoreless draw or a win in the second leg, to be played next Wednesday night at home at Olympic Stadium, to become the first Canadian team to win the CONCACAF tournament. America needs a victory or a tie by more than two goals to win its fifth CONCACAF championship and first one since 2006.
"Despite not getting a win, the tie gives me good vibrations because it's hard to generate football and we had like 12 or 13 scoring chances today," America coach Gustavo Matosas said. "Fortunately, for us Oribe came through for us. Montreal is a good team, that's why they are at the final, but I'm confident that we can get a win at their place."
Impact goalkeeper Evan Bush faced 28 shots, including nine on target. Moises Munoz Rodriguez faced three Montreal shots.
The Impact, the first Canadian team to appear in a CONCACAF final, is trying to become the first non-Mexican team to win the tournament since Costa Rica's Saprissa in 2005.
"A 1-0 win would have been great, but I think a 1-1 draw gives us confidence for the second leg," said Montreal coach Frank Klopas.
With files from The Canadian Press