Hill said: "We knew that we were well rested having rested most of the players from the Leicester game.
"Bath had a tough game against Sale so we knew that come the last twenty minutes we would have the strength and the legs to keep going.
"So going into the game on New Year's Day at Sale we should have the advantage as the Sharks look tired at the moment.
"They have had some massive games in Europe, big games against Bath and Newcastle, and at the moment their players look on their knees a little with their injuries."
Bath's worries continue as they remain in the bottom half of the table at the midway point in the season.
New head coach Steve Meehan said: "The defeat is disappointing for the players coming after the Sale victory.
We felt that we showed that we were maturing as a side but in this game we couldn't capitalise on anything that we created."
Saracens director of rugby Alan Gaffney hailed his side's 38-15 win over Northampton which moved the London club up to fourth.
Gaffney made it two wins on the bounce over Christmas after Saracens scored 22 unanswered points after the break.
"It has been good - we've had a great Christmas period," he said. "Nine points in two games - I'd have taken that before."
Northampton coach Paul Grayson was left to reflect on an injury crisis that has seen the Saints struggle with the hectic Christmas schedule.
Grayson said: "We lost our shape after the break. We were able to kick for territory in the first half and put Sarries under pressure but in the second half we were hanging on as we were going backwards."
Worcester coach John Brain has urged his team not to "fragment" under pressure as they battle against relegation.
A 27-20 defeat at the hands of Harlequins left the Warriors nine points adrift of Northampton at the foot of the table and with only one win in the league this season.
"We'll be totally written off now and we've got nothing to lose," said Brain.
"The test is what happens over the next few weeks. As a team we can either fragment or we can stay together."
Harlequins are now eighth, 12 points clear of Worcester and rugby director Dean Richards believes they can climb further away from the relegation scrap by beating the Falcons.
"The Newcastle game on New Year's Day is now a massive one for us - I think they would have been expecting us to lose and then to come and beat us," said former England number eight Richards."