After a goalless 90 minutes, Francois Zoko rifled the Glovers ahead early in extra time but Akwasi Asante levelled from the spot to send the replay to penalties.
Ryan Dickson and Tom Eaves suffered shootout heartbreak while Solihull scored four from four to progress to the second round for the first time in their nine-year history.
“I’m disappointed and I want the players to be disappointed and hurting too because it’s not nice getting knocked out of the FA Cup,” Way told YTFC’s official YouTube account.
“We’ve been quite strong recently and I’m not normally critical of the group but that wasn’t up to our standard.
“I’ve been involved in many FA Cup games to know it was going to be a difficult task and wasn’t going to be easy.
“We knew we were going to be tested and they had no fear. I have to respect them for that and say well done to them.
“It’s my first ever defeat as a manager where it’s been a banana skin so I have to analyse and evaluate but I’m quite confident the group will respond and I’ll make sure of that.”
Yeovil return to Sky Bet League Two action against Colchester United on Saturday, November 19th at Huish Park (3pm).
The Essex club are winless in their last nine league encounters, while Town could move into the top seven with victory.
“I have to see a group of players that look hungry for Colchester because that’s what Solihull had,” Way said.
“I have to make sure I get them going again as we’ve got an opportunity to go into the play-offs.
“What I’m asking from the players is for a very good performance to put this disappointment to bed.”
Fans can purchase tickets for Saturday’s match online or by phone on 01935 847888.