Three fourth-quarter interceptions from the rookie quarterback allowed the Patriots to hold on for a narrow win at the Gillette Stadium.
The 22-year-old told the team's website: "I think we came into it and had a great plan.
"We knew their defense plays well as they always do and it seems like every week I'm standing here saying if I don't turn the ball over, we win games, and ultimately that's what it comes down to.
"In this league, no matter the yards you get, no matter the time of possession, it's about the turnover battle and I cost us the game."
Smith completed 214 passing yards and was sacked four times during the game.