Fati, 17, became the youngest goalscorer in the competition's history, netting a late winner in a 2-1 victory at San Siro on Tuesday.
The attacker has three goals this season and Valverde praised the Spain youth international, whose strike came after Romelu Lukaku had cancelled out Carles Perez's opener in the Group F clash.
Barcelona's 17-year-old Ansu Fati waves goodbye to Inter's Champions League season ✨ pic.twitter.com/joLITbWZJx— Bleacher Report Live (@brlive) December 10, 2019
"They [Fati and Perez] are two players who have reached the first team this year. Young players, very young in the case of Ansu," Valverde told a news conference, via AS.
"The game was complicated, especially for Carles Perez because they had an experienced defence and he started, but he has a goal because he hits the ball well.
"Ansu has been going through some physical difficulty, with some discomfort, but he is a born scorer."
Barcelona had already booked their place in the last 16 and they finished top of the group as Inter were eliminated.
Valverde talked up the importance of the Champions League to the Catalans, who last lifted the trophy in 2015.
"We have a very big obligation with our club and with this competition," he said.
"We have shown that you have to win everywhere and today's game was not going to be an exception."
After criticism earlier in the season, Barcelona top LaLiga and are on a six-match winning run in all competitions.
But Valverde wants to see more from his team, who make the trip to Real Sociedad on Saturday.
"We are fine, but I like to see everything in a more continuous way. At first we had difficulties," he said.
"This group phase has been difficult for us. We suffered to overcome Inter, to win at Slavia [Prague]. But we are here and we have a tremendous challenge with three games in a week and we want to close well."