Saban pointed out the various mistakes his team made, whether it was busted coverages, a blocked punt or a penalty for roughing the kicker.
Wide receiver Jameson Williams had a team-high 123 yards, but he also fumbled in the red zone early in the game.
"We shot ourselves in the foot," Saban said. "I think those things need to get corrected or they're going to cost us at some point."
Saban said his team must use its bye week wisely.
"We have a real team we have to play in two weeks," he said, referring to LSU, Alabama's opponent on Nov. 6. "We need to get some rest. We need to recover. We also need to focus on what we need to do to finish the season and get some of these things corrected."
Saban said that ultimately the difference against Tennessee was Alabama's ability to control the ball, dominating the time of possession 40:26 to 19:34.
Early on, it was quarterback Bryce Young making plays with his feet, rushing for 42 yards and two touchdowns.
Running back Brian Robinson found his groove in the second half and wound up with 103 yards and three touchdowns on the ground.
Saban said that while everything wasn't perfect, "we made the plays we needed to win the game."
"I'm not going to be negative about the game," Saban said. "I'm not going to be negative about our team."
Case in point: On the one hand, Saban was disappointed with the defensive secondary, which gave up several big plays through the air, including a second-half touchdown in which cornerback Josh Jobe wasn't lined up properly at the snap.
But on the other hand, Saban said, "The offense answered and we kept scoring points."
Saban credited Young with several "critical" scrambles to keep drives alive. The sophomore quarterback completed 31 of 43 passes for 371 yards and two touchdowns.
Alabama's 52 points was the most it has scored against a Power 5 opponent this season. It was also the Crimson Tide's 15th straight win over the rival Volunteers.