Bryan Mbeumo had Brentford ahead before Gabriel Jesus, Mikel Merino and Gabriel Jesus struck for the visitors.
Frank said afterwards: "You need to play a perfect game against Arsenal, one of the two best teams in the league, probably in the world. I thought actually we were close to perfect in the first half, I loved the way we played.
"I'm pleased with the performance, especially the first half, and also I thought the second half was not too bad - we just conceded two goals.
"I think the overall performance was good, maybe very good, but the big moments we didn't take them well enough today - especially the three defensive moments.
"Of course I would love to have a full squad, but I haven't so we'll find solutions for that. We'll deal with it. The medical department is working extremely hard so we'll find a way."
On Arsenal's title challenge, Frank added: "It's a long season. I understand why Arne Slot is saying 'we haven't won anything yet'. Liverpool haven't had a dip yet and that will most likely come, and Arsenal are picking up points when they're not fully firing. It will go right to the end."