I could be wrong but I think the only way to get a 1-point safety is on a PAT attempt. This means you need to score a TD first, go for 2, turn the ball over to the other team, have the other team run into the end zone and you then tackle the other team for a 1-pt safety. End result you get 7 points total. 6 for the TD and 1 for the safety.
Now I suppose its possible the team kicking the extra point could run backwards into the other teams endzone and gets tackled for 1-pt safety. So your team gets 6 points and the other team gets 1 point. So it is possible for 1 team to score only 1 point in a game, but in order for a team to score only 1 point, the other team has to score a 6-pt TD first. So its impossible to win a game 1-0.
So, yes the minimum number of points it takes to win a game is 2 points.