Hill's men produced a massive surge in the opening ten minutes - as Bath were reduced to 14 men with Dave Attwood in the sin bin - and saw Aleki Lutui held up as they camped on the Bath line.
However, the home side held firm and with the huge confidence boost of keeping Warriors out they duly ran away with the game in the closing stages.
"I'm disappointed because a lot of effort went in," said Hill, who later revealed an illness bug had hit the camp before kick-off and forced Andy Goode out.
"For the first 40 minutes I felt we took the sting out of Bath and then we had good pressure at the start of the second half.
"We had a great chance to score from the driving lineout and then the scrum. That was the turning point and all the effort, around six or seven minutes of sustained pressure, came to nothing.
"If we had scored then we would have gone into the lead and that would have given us a huge boost in confidence. Bath would have had to look hard at themselves.
"Not only did we not get over then, we conceded a penalty and from the lineout they score.
"It was a killer blow and knocked the stuffing out of everyone. If we had scored then we had a chance of beating Bath, they weren't on top form, but once they got that score their confidence soared and it was a hard afternoon.
"The comments in the dressing room at half-time were that we had made mistakes but we were working so hard to cover them up. We were under the pump a little bit but the will and energy was superb.
"We just needed that score and when you are in our part of the table one score makes a huge difference to the confidence.
"You could see it was the game changing moment, we had to score then. Psychologically as a defence if we can defend on your own line for several minutes and you come away with a penalty then the swing in confidence is massive. You could almost see the game had gone."
Warriors also lost three players to the sin bin at The Rec as Dean Schofield saw yellow in the first half before Craig Gillies and David Lemi were sin binned in a franctic finale as Worcester were forced to play for 13 men.
"The yellow cards were down to indiscipline and a lit bit of misfortune with Lemi," added Hill.
"Craig conceded a silly one and he apologised to everyone at the end. A yellow card at the end, when you are still within two scores of getting back into it, was not needed.
"Bath have a lot of quality in the team and when they want to play they are a very good team. When they got their confidence up and started to throw it around it can be tough to stop."