Sarries had the chance to go top of the Aviva Premiership table had things gone their way but four wayward penalties, by Nils Mordt and Sam Stanley, ultimately cost them dear.
As it was, Andy Goode secured the win for Worcester with three penalties and a drop goal to balance against a solitary Mordt penalty.
Hill said: "What a performance. We were full of intensity, commitment and passion for the whole of the 80 minutes. We have been feeling a bit of pressure as we felt as a club that we had gone into our shell over the last three weeks but from the kick-off tonight we were terrific."
Despite Hill's contentment, he believes the margin of victory could have been even larger after his side failed to cross the tryline.
"We should have won the game by a lot more than 12-3," he said.
"We are getting into the opposition 22 but failing to finish off the good approach work. It makes a big difference if you get one try as it pushes you away from the opposition."
Worcester lost young centre Joe Carlisle with a leg injury early on, leaving Hill sweating on any future lay-off.
"I was disappointed to see Joe come off because I wanted to see how the 10-12 combination went with Andy Goode but unfortunately he only lasted 15 minutes," Hill added.
"I believe he has strained ligaments on the outside of his ankle but hopefully it's not too serious."
Saracens director of rugby Mark McCall was an unhappy man after his side struggled to create try-scoring opportunities and botched a series of presentable kicks.
"It was a dreadful game of rugby, I feel sorry for anybody who had to watch that," he said.
"It was a very, very wet pitch but that's no real excuse, there was no flow to the game and it was very depressing."
"We made a lot of unforced errors and were regularly penalised in the scrum area that cost us field positions. When they got into our territory, they kicked their penalties which we didn't."
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