Miles Addison punished a mistake by goalkeeper Romain Larrieu to thump a header into the net and win Saturday's Championship contest at Pride Park.
Larrieu had told his defenders to leave Gary Teale's corner to him but then left the ball and Addison undid all Argyle's hard work.
Sturrock explained: "Romain called for it, he said he was going to come for the ball so the players left it to him and the next thing it's in the back of the net.
"I am gutted because the players gave everything they had today.
"It looked as if our hard work and desire was going to get us a result, but it's a strange game and it can kick you in the teeth at times.
"I thought we had weathered the storm, in fact a minute before the goal we should have scored at the other end.
"We have now got seven days to wait until our next game but I wish we had one tomorrow to get this frustration out."
Derby manager Nigel Clough was delighted with the way his team kept going to get a late winner for the second home game in a row.
The Rams scored towards the end of their opening game of the season against Peterborough and Clough admitted: "It's important at home that you get results.
"I thought we deserved to win on the balance of play. We were a little bit tentative early on for some reason but when we got going, we dominated possession."
Clough has called for goals from all over the pitch and Addison has responded with two already.
He said: "I think he's our joint leading scorer now! The two centre-halves are a threat and we put good balls into the box and looked pretty dangerous from set plays.
"I thought we were dangerous from corners and free-kicks against a side that are good themselves at set-pieces."
Derby celebrations were muted by what looked like a serious injury to striker Steve Davies, who was stretchered off early in the second half after falling awkwardly in the box.
Clough said: "Steve is strapped from thigh to ankle but we don't know what the problem is at the moment.
"It could be a knee, ankle or both."
Derby had started slowly and Argyle deserved to be in front before Alan Judge was brought down by Jake Buxton after Jamie Mackie had struck a post.
Judge easily beat Stephen Bywater with the spot-kick but Derby hit back and were level six minutes before the break.
After Davies had seen a header saved, Robbie Savage volleyed the ball back in and Buxton turned it into the net.
Derby were far more urgent in the second half and put Argyle under sustained pressure, which paid off in the last minute when Teale delivered a corner to the back post and Addison did the rest.