The Glovers fell behind to a first-half header from John O’Sullivan, before Ben Whitfield saw red in the second period for an alleged elbow on Tom Miller.
Less than a minute later, the play-off chasing Cumbrians doubled their advantage as Reggie Lambe lashed home at the second attempt.
Way says himself, his staff and players have to take “collective responsibility” and will do everything to bounce back at Portsmouth next week.
“I’m not going to turn my back on them,” he told YTFC’s official YouTube channel.
“I’m going to stay resilient and give them confidence going into the next game because it isn’t going to be easy between now and the end of the season.
“It’s my responsibility to make sure their spirits are high going into the next game.”
On his team’s performance, Way added: “We had large spells of possession before their first goal went in.
“They put men back behind the ball and we couldn’t break them down with any clinical play. Then there were one or two missed passes and the crowd weren’t happy.
“We had to change from 4-4-2 and I was going to change to 3-4-3 but then there was the sending off and we lost a number in there. I’m disappointed because at that stage of the game the last thing you need is someone to get sent off.
“There was play where we lacked a little bit of quality. We have to regroup and have a really busy week next week.”