Sturrock made four changes to the side beaten by West Brom at the weekend, with Chris Clark, Cillian Sheridan, Krisztian Timar and David McNamee coming in.
Watford named an unchanged line-up following their 1-0 win against Barnsley last time out.
The struggling Pilgrims found themselves a goal down after just four minutes.
Don Cowie's corner found its way back to him and he picked out Cleverley who swept the ball home from six yards via Gary Sawyer's outstretched boot for his third goal of the campaign.
Goalkeeper Scott Loach came to the Hornets' rescue after 15 minutes, pulling off a stunning save to tip over a powerful header from Sheridan.
Loach denied Jamie Mackie shortly afterwards after good play by Clark, before a Clark corner from the right was headed wide by Timar as Argyle pushed for an equaliser.
Romain Larrieu was called into action for the first time after 40 minutes when he comfortably gathered a header from Danny Graham.
But there was nothing he could do three minutes later when Lloyd Doyley played a free-kick to Henri Lansbury and he let fly from 20 yards. The French keeper was beaten but the ball cannoned off the woodwork.
Sheridan brought another fine save from Loach with a close-range header at the start of the second half, while Kari Arnason and Mackie both sent headers wide.
The home side were dominating, with another Clark corner causing problems, this time Sawyer powered a header straight at Loach who palmed over.
Argyle came agonisingly close to the goal their pressure deserved on the hour when a Karl Duguid header came back off a post with Loach stranded.
Sturrock made three changes in the 73rd minute, bringing on Bradley Wright-Phillips, Alan Judge and Rory Fallon in a desperate attempt to get something from the game.
They came close through a Duguid piledriver, while Carl Fletcher and Timar blazed wide but Watford held on.