Jordan Cook’s 11th-minute strike and a superb Alan Sheehan free kick early in the second half had the Hatters in control before a topsy-turvy, eight-minute goal frenzy ensued.
Ivorian forward Francois Zoko reduced the deficit just shy of the hour mark, but Isaac Vassell restored Luton’s two-goal advantage moments later.
The gap was back to one when Omar Sowunmi nodded home Town’s second of the night, however the Glovers’ desire for a quick equaliser was short-lived as Danny Hylton made it 4-2 to the hosts within 60 seconds of the restart.
Vassell put extra gloss on the scoreline with a fifth in injury time, as Luton reached the semi-finals of the competition for the first time since they won it in 2009.
“There are lessons to be learned,” Way told YTFC’s official YouTube channel.
“Certain individuals would’ve taken a lot from the opponents we’ve played tonight. They will know certain players caused them a lot of problems and they need to reflect on that.
“The players are very, very disappointed. They were tested tonight but I can’t fault their effort and commitment.
“We have got a small, young squad and we’re learning.”
He added: “I can take a lot from it (the Checkatrade Trophy) and I feel the players have taken a lot from it.
“We’ve had a good journey with some really good games and we’ve played in some good grounds.
“I think the players will say they’ve enjoyed it. We’re disappointed we’ve been knocked out but they’ve given everything in this trophy.
“The effort and the energy the players have shown in this cup is something I’m very proud of.”