After striking the post in a goalless first half, Town conceded twice in the second period to eventually lose 2-0 at the LCI Rail Stadium.
Tin Plavotic and Billy Waters were on target for the hosts, getting revenge for the 4-2 drubbing the Robins suffered at Huish Park in the reverse fixture.
Speaking to YTFC’s official YouTube channel, Way took responsibility for his side’s performance.
“Our supporters are emotional,” he said. “If I was a supporter I would be disgusted with that second half performance.
“We didn’t get in the final third enough, skipped tackled and arrived late. They won the majority of second balls in the second half which wasn’t the case in the first half.
“I have to make sure I analyse myself and I will do. I’m a character that will show fight and determination.
“This is a special Football Club and Yeovil got to where we are with passion and determination and I didn’t see that in the team today and I have to take responsibility for that.”
Two set-pieces meant Yeovil left Gloucestershire empty handed, with Plavotic nodding home from a free-kick and Waters from the penalty spot.
Aware of their dead-ball threat before the game, Way was disappointed with the goals conceded.
He said: “We knew it was going to be a set-play game with the way they play. They have towels for wiping down balls, it’s slow, but you have to deal with that.
“We didn’t, and going 1-0 down so soon (after starting the second half) was disappointing.
“You have to give them credit, they worked hard. But our players are playing for their futures, all of us are.
“I will take responsibility of the group; the way I manage it, the formation, I take all of that. I knew the day I became manager that it’s my job on the line.”