Reiss Nelson scored what proved to be the winner only eight minutes into the all-Premier League affair in west London but the Gunners did not have it all their own way.
After a disappointing 3-1 loss to Everton on Saturday, Frank had vowed his team would bounce back.
While the Bees offered little in the opening 45 minutes, the hosts improved after the break, with Yoane Wissa forcing a stunning save from Aaron Ramsdale in the second-half.
Keane Lewis-Potter had an effort cleared off the line in the dying embers too but, ultimately, Arsenal held on.
Bees boss Frank said: “I hate losing so it’s always tough to take but very pleased with the performance. That is the thing we can build on.
“Good performance in the first half and excellent in the second half. I think Arsenal were very lucky to get away with this with the chances we created in the second half.”