The Chilean scored in extra time as the Gunners came from behind to beat Manchester City at Wembley and reach a third FA Cup final in four years.
Amid ongoing speculation over Sanchez's future at Arsenal - the club have offered their star player a staggering £300,000-a-week contract offer to shatter their wage structure - Wenger was full of praise for the goalscorer.
"Alexis Sanchez was like the team," he told BBC Sport. "He had problems to start and became stronger and stronger. He is an animal, always ready to kill the opponent. He will never give up. He will be here next year because he has a contract and hopefully we will manage to extend him.
"When at Arsenal football club we want the team to turn up in big games and I think we done that today. Overall I am very pleased. People questioned us, we went through tough times. In these times you can be divided or united and today we have shown the right response."
Sergio Aguero had given City the lead only for Nacho Monreal to volley home a rare goal to take the game to extra time before Sanchez's winner.
Wenger was thrilled with his players' response to conceding first, adding: "It was a big game today. Overall it was a great performance not only technically but mentally.
"We became stronger and in the end deserved to win the game.
"The team performance was there. When the players give their best and are completely focused you always stand up for them.
"We were close to scoring and on the break they are very good. We lost the ball and conceded the goal but we showed the right response."