The Thorns have won the league's first two titles and lifted the trophy again in the last year.
Smith scored in the 4th minute after the Current had a costly turnover.
Smith had a chance to get another goal in the 27th minute. Lynn Williams scored in the championship game for North Carolina and she is the only other winner of the award. She scored in the league final at the age of 22.
Morgan Weaver had a shot from the edge of the box but Franch got a hand on it before it hit the bar.
The Portland Thorns, one of the NWSL's teams, was at the center of the Yates report that showed systemic abuse and mismanagement.
The game, which was broadcast in prime time on network television for the first time, was almost sold out, with some fans holding up a sign that said "support the players."
In the playoffs, Portland beat the San Diego Wave in the semifinals and advanced to the championship game.
Smith, a member of the U.S. national team, scored 15 goals for the Thorns over the course of the regular season and playoffs and was also named the game's Most Valuable Player.
Many of the players from the Utah Royals came over to play for The Current. They were last in the league.
The Washington Spirit won the championship.
Information from the AP was used.