The 24-year-old joins from the professional National League North Aggborough outfit on an initial one-year deal with the option of another year.
He was out of contract at Harriers and had been offered revised terms.
Gnahoua, of French and Ivory Coast descent, has previously had spells with Stalybridge Celtic before moving on to Macclesfield Town.
He has also played for Spanish side CF Gava.
He was Harriers' top scorer last season, finishing with 13 goals – 12 in the league.
Gnahoua, who joined Harriers in January 2016, netted in the Aggborough club's play-off semi-final, a screamer from outside the area – but the side eventually lost out to Chorley on aggregate.
He becomes Paul Hurst's first summer signing with Shrewsbury.
Hurst said: "I'm pleased to get him in. He's a player we've watched quite a few times.
"He's got plenty of attacking ability and one we feel, is not a gamble, but worth exploring further.
"Hopefully he can make the step up as quickly as possible."
Gnahoua, who was a favourite with Kidderminster fans across his 16-month stay, was delighted to seal the deal that sees him rise up three divisions.
"I'm definitely pleased to be here, playing in League One, coming from a lower league it's going to be a good experience for me" he said.
"I'm delighted. I hope I can do as well as I did last season and the manager gives me a chance.
"Unfortunately we didn't get promoted but it was a good team and a good experience for me."