Griezmann made his first LaLiga appearance for Barca, which suffered a shock 1-0 defeat to Athletic Bilbao in its opening fixture on Saturday (AEST).
A €120 million ($198.5 million) signing from Atletico Madrid, all eyes were on Griezmann, but he struggled to make an impact in absence of captain Lionel Messi, while Luis Suarez's first-half injury left the Frenchman somewhat isolated.
"If he's near the area, he is decisive and moves well, but you have to dominate the game for that to happen," Valverde said.
"Putting him on the left side, we know that you are going to focus on the goal, but we have a side and players that can hold the game.
"Of course, he is aware that we all expect a lot from many players and have to seek to get more into the game."
Suarez left the game just before the break because of an apparent calf injury, which came moments before he missed a chance to put Barca ahead by firing a shot agains the post.
Valverde added: "He has a muscular problem and we have to wait for the tests. In the second half, when we were around Athletic's area, we missed him."
Barca eventually succumbed to Athletic, which triumphed thanks to veteran Aritz Aduriz's stunning overhead kick in the 89th minute.
▶️ WATCH it again and again 🙌🏼 #AritzAduriz's OUTRAGEOUS winner is the talk of football today!! All the highlights from @AthleticClub 🆚 @FCBarcelona are here 👉🏻 https://t.co/jaq77elFNG #LaLiga #AthleticBarça pic.twitter.com/uD7kAed20V— beIN SPORTS (@beINSPORTS_AUS) August 16, 2019
The 38-year-old made an immediate impact off the bench, meeting Ander Capa's cross in sensational fashion to seal all three points.
"With Aduriz, any ball in the air is a danger and he got his goal," former Athletic boss Valverde said.
"That move is dominant, and he's scored similar goals. When I saw him line it up... I wasn't surprised with the execution at all."