An awful defensive mix-up in the Orient back line saw Gaby Zakauni concede possession with a tame header and his central defensive partner John Mackie tripped David Edwards as he surged towards goal.
Referee Ian Williamson pointed to the spot and McMenamin fired the ball home to the left of Orient goalkeeper Glyn Garner.
After that the visitors were forced to dig in and fight a rearguard action as Orient threw everything at them but they couldn't find a way to breach the solid defence.
Orient forced eight corners in the first half as they piled on the pressure and skipper Gary Alexander saw his early header cleared of the line by Town's impressive Jamie Tolley.
But for all Orient's pressure, clear-cut chances were thin on the ground and it was Shrewsbury who threatened to extend their advantage on the break.
Home keeper Garner had to tip a Kelvin Langmead shot away and then he saved a 20-yard blast from Edwards before he pulled off another smart stop when he tipped away a Tolley free-kick.
But Orient ended the first half on the attack and Joe Keith had a close-range shot blocked.
The second half was played almost exclusively in the Shrewsbury half, but with skipper David Walton and Richard Hope forming a rock-solid barrier, Orient struggled to make their pressure count.
Alexander had a header well saved by Town keeper Joe Hart and 14 minutes from time Orient's Efe Echanomi and Donny Barnard both had shots blocked.
As the game progressed Orient manager Martin Ling threw caution to the wind and pushed more players forward, but it was Town who looked likely to stretch their advantage on the break.
Indeed, seven minutes from time Orient's Barnard was caught in possession and McMenamin raced clear but with only keeper Garner to beat he dragged his shot wide of the target.