The 27-year-old Slovakian shot-stopper checks into Montgomery Waters Meadow on an undisclosed fee from the Ricoh Arena, where he had another year left on his contract.
Marosi has enjoyed a very impressive spell with the Sky Blues since he checked in there on a three-year deal from Doncaster.
Shrews boss Steve Cotterill revealed, after a testing period of negotiation, he has secured the goalkeeper he has been searching for. Coventry are set to add to their goalkeeping department this summer and Marosi was made available for transfer.
The boss highlighted Town's new goalkeeper's quality with the ball at his feet, as well as his shot-stopping ability, as part of his strengths.
Cotterill said: "Marko has been a goalkeeper I've wanted to sign the minute I knew we could get him from Coventry. It has been quite a long process getting to where we are, but we are delighted to have signed Marko.
"He will bring great attributes to the squad. He's a great footballer for a goalkeeper, an excellent shot-stopper and he's quick off the floor."
He was named in the League One team of the season in his debut campaign, 2019/20, when the Sky Blues won the title and promotion to the Championship.
And he kept his place as No.1 in the second tier last season, in which he made 21 appearances in all competitions aside from suffering a horrific facial injury, where he fractured his eye socket and cheekbone last November.
Shrews boss Cotterill is continuing to build on an ambitious transfer period in his first summer window as Town boss.
He has been in the hunt for a permanent No.1 and Marosi has been handed a vote of confidence in the race to be first choice next season, having been handed a lengthy three-year contract after Shrews parted with a fee to bring him away from City.
The manager alluded to how Shrewsbury have been 'lucky' in their recent success with loan goalkeeper, but Cotterill was keen to bring in a permanent addition.
Cotterill added: "We are delighted that he's signed permanently, and that's something I've wanted to do for a long time. We have been lucky with our loan keepers but having our own is fantastic.
"It has been hard work to get it done but these deals don't happen overnight. Now we've got Marko in the bag, we'll move on to the next."
Marosi also enjoyed an impressive full season in League One at previous club Doncaster before switching to the Sky Blues.
He will compete with Harry Burgoyne for Town's new No.1 next season.
Marosi is Town's fifth new signing of the summer, including former loan defender Matthew Pennington, left-back Luke Leahy, versatile former Blackburn club captain Elliott Bennett and striker Ryan Bowman.
The boss is pressing on with a number of deals as Town continue an ambitious push in the market, with a move for a right-back in the offing, as well as reinforcements in defence, central midfield and up front.