Barca's 19-year-old prospect rescued his team in the 70th minute with a fierce volley from inside the area after the home defence failed to clear Oscar Mingueza's cross from the right.
However, the Ukraine champions gave Barcelona a hard time in a frantic game, with both teams missing several chances.
It was Barca's first away win in all competitions this season.
"We all knew that we had to win at any cost. The players were nervous before the game and you look at them now and their faces are different. Winning always helps and I hope we keep momentum for the next game," Barcelona's caretaker coach Sergi Barjuan told reporters.
After losing Sergio Aguero for three months following a cardiac evaluation, Barca welcomed back Fati and Ousmane Dembele from injury and both showed, alongside fellow forward Memphis Depay, that they can bring the spark the team desperately need.
After dominating the first encounter at Camp Nou and winning 1-0, Barcelona faced a much-improved Kiev at their home ground.
Playing with a deep defensive line and relying on counter-attacks, Dynamo threatened the visitors goal and came close to opening the scoring at least three times in the first half.
Ter Stegen made a critical save at the end of the first half after a powerful shot from outside the box by Carlos de Pena.
Barcelona were also probing, with Depay and Fati involved in their attacks, but only managed to get close to Dynamo's goal in the second half when Dembele came off the bench.
Referee Ovidiu Hategan awarded Barca a penalty for a foul on Fati but changed his decision after a VAR review. However, it was not long before Fati scored the winner with his close-range blast and Barcelona managed to see out the victory.