Over 800 Green and Whites travelled to St James Park as Town had to settle for a point after being 3-0 up.
The Glovers return to Huish Park this weekend and Way is craving three points for the fans.
“We have to apologise to the supporters,” Way told YTFC.net.
“As a manager, setbacks come hard and fast. I don’t think they come any harder or faster than they did against Exeter.
“Saturday is a game in front of our home supporters to be able to put on a performance.
“I’ve tried to get the players to think like winners this week as it’s for important for them to become winners on Saturday. It’s also important for the supporters as well because they deserve that.
“All I can ask is for the supporters to get behind the players because we have got some younger players and we all want to finish as strong as we can, in the highest position we can. I think the crowd are going to play a big part in that.”
Yeovil take on an out of sorts Carlisle outfit who dropped out of the Sky Bet League Two play-offs for the first time since September following a 2-0 defeat to Crewe Alexandra.
The Cumbrians are winless and goalless in seven matches – losing six of those – and Way is challenging his players to deal with their physical and direct approach.
“They’ve had a poor run of form,” he said.
“We know what they’re all about. I’ve played against Keith Curle’s teams for a number of years.
“They’re very, very direct. It’s going mean a lot to them and I’m certainly expecting another physical tussle like we’ve had against his sides over the years and the players have to deal with that.”
Tickets for the match can be purchased in person at the ticket office, online or by phone on 01935 847888.