Worcester had seen victory snatched away in previous home games against Wasps and Harlequins by late scores and were forced to settle for a draw at Newcastle in similar scenes.
But this time Hill's men showed they could close a game out as a late drop goal from Andy Goode put the gloss on a crucial league win.
"We have managed to close a game out and that is a key thing for us," admitted Hill.
"We have had games here against Wasps and Harlequins that we have just lost out to, we went to Newcastle and were leading right until the end and only drew the game.
"Teams were looking at us and thinking if they could hang in there with ten minutes to go then we would crack and they would score.
"With ten minutes on the board it was a big psychological thing for us to close the game out, get that drop goal and give us a cushion.
"It was a massive step forward," he added. "The players are working very hard and the attitude is superb, we have worked really hard on the attack over the last few weeks.
"But if you keep losing by the odd point then there comes a time, if you are not careful, when suddenly those little loses by a point become a bit wider.
"To get that win will give everyone a big boost. That is what we need at this time of year, to get a win like that shows we can play well and close the game out.
"We asked the players for our best performance this season and I think we did that.
"The first half belonged to us, probably 35 minutes out of the 40, and we had a five minute spell when we had to defend hard and eventually got a turnover.
"Bath were hit by their lineout, although I thought Mark Lilley, who is probably fourth choice for them and is a prop, did well in the circumstances. That helped us a little bit."
Hill also praised the attitude of his players as they looked to adopt an attacking style and were never afraid to get hands on ball.
"We started with good ball retention and what I liked was the way we played, we didn't look jittery and everyone looked comfortable on the ball," he said.
"The forwards and backs were mixing and, although I think we have been criticised for a lack of attack, we went out with confidence.
"We didn't get the rewards until right on the stroke of half-time when Miles conquered up a superb try. We deserved to go in 10-0 up.
"We tried a different tactic tonight where we hit the ball up the middle and then straight to the backs.
"In the past we have taken the forwards on, but we decided with the runners we have got we wanted to get the ball in their hands a bit more and that worked for us."