The Hornets made it six matches unbeaten thanks to Tom Cleverley's fourth-minute winner in Tuesday night's clash at Home Park, moving into the top six in the process.
However, the narrow defeat was Argyle's fifth in succession and Sturrock said: "I am heartfelt sorry for the team.
"We desperately need a win and next up we go to Newcastle, so I am sure everyone will have them down for a home win.
"I could see that we have still got a spirit within the team - and that's vital.
"Their keeper has made half a dozen gilt-edged saves - and Karl Duguid has hit the post. That tells its own story without me having to say anything.
"We are creating chances so that gives me real hope.
"I can see the positives are there and now we need a settled back four and hopefully that will stop the nerves at the back, because up front we are looking all right and I am sure the goals will come."
Watford opened the scoring from Don Cowie's corner from the right.
Cleverley swept the ball home from the edge of the area with a powerful first time strike that was deflected off a defender as it flew past outstretched home goalkeeper Romain Larrieu and into the top corner.
Watford forced 10 corners in the opening 20 minutes but Argyle did manage to counter and force visiting keeper Scott Loach into action.
Loach made a diving one-handed save to claw Jamie Mackie's header off the line in the 14th minute after Cillian Sheridan headed on Chris Clark's free kick.
Sheridan was also denied by Loach, who made a brilliant point-blank save to deny the on-loan Celtic striker in the 20th minute from another Clark cross.
Left-back Gary Sawyer's 37th-minute angled volley also took a deflection, off striker Mackie, and flew wide of goal.
Larrieu was forced into action in the 40th minute as Cowie's cutback was met by Danny Graham's first-time strike, forcing the home keeper to save on his line.
Loach made a routine stop to keep out Duguid's 48th-minute effort but made a breathtaking stop two minutes later to again deny Sheridan.
The recalled striker spun on his own header to let fly with a volley that Loach did brilliantly to push over the bar.
After Kari Arnason flashed a header inches wide, Loach flapped Sawyer's header over the bar and then Sheridan volleyed wide, while Duguid then crashed a header off the post.
Sturrock made a triple substitution in the 73rd minute introducing fit-again summer signing Bradley Wright-Phillips for his Pilgrims debut, winger Alan Judge and forward Rory Fallon.
Fallon's diving header produced another magnificent, full-stretch save from Loach, who then palmed over Duguid's dipping volley.
Watford boss Malky Mackay said: "I am absolutely delighted with the fact that we came down here and have come away with three points, a clean sheet and a battling performance.
"Plymouth threw everything at us in the second half and a really young squad showed immense heart, courage, team effort and maturity to actually hold out and create at the other end, which I was absolutely delighted about.
"I thought in the first half an hour we were immense but perhaps should have dominated more with chances."