A thumping header from James Yorwerth and a penalty from James Collins was enough to condemn the Glovers to defeat in their first match of 2017.
Town are now winless in five Sky Bet League Two matches and Way wants to see more from his players.
“The group has to take responsibility,” he told YTFC’s official YouTube channel.
“If you’re a forward you get paid to score goals, if you’re a defender you have to make sure you defend well and keep the ball out of the back of the net – that applies to the goalkeeper as well.
“Coming to a place like Crawley, could they show me a little more knowing what we were up against?
“I didn’t see enough of that responsibility and I was a little bit disappointed with the experienced players.
“We knew it wasn’t going to be easy but I didn’t see enough fight, cleverness or spirit and that’s one thing we’ve had to get us to where we are now.”
Town are back in action next Tuesday (January 10th, 7pm) when Reading Under-23s visit Huish Park in the Checkatrade Trophy.
Tickets can be purchased in person at the ticket office, online or by phone on 01935 847888.