Tigers scored two tries in the opening hour of Friday's cup clash at Welford Road before Billy Twelvetrees collected his second of the evening with just five minutes remaining and then Tuilagi collected an interception score with the final play.
That sealed a 27-11 victory, maximum points and top spot in Pool 1 with one game to play.
Cockerill said: “There were a lot of guys out there who had not had much game-time recently and, understandably, there was a bit of rustiness in the combinations.
“But the players had a great attitude, they kept playing and eventually got the breakthrough.
“The interception try at the end brought the bonus point and that keeps us alive in the tournament.”
Twelvetrees collected his opening score on the stroke of half-time, latching on to a chip through by captain Aaron Mauger, then added a second in the 75th minute to kill off any lingering hopes of a Bath fightback.
“Billy is a real prospect, he has a natural talent and just needs to keep refining that,” said Cockerill.
“Early in the season he couldn’t play the second-team games because we couldn’t risk him with so many injuries in the squad, then he got an injury himself. We’ve just got to make sure his development is done right and we do that pretty well on the whole, there is lots of evidence to back that up.”
Victory took Tigers level on points with Saracens in Pool 1, with the teams separated by a try count of 9-6 in Leicester’s favour, and one game to play.
“Saracens are at home to the Dragons next week, while we will do what we can at Northampton,” Cockerill said.
“Northampton are on a good run and had a good win over Sale tonight. It’s a local derby and that adds a bit of spice to it too.”