Leicester Tigers made it maximum points from the opening two fixtures in Pool 5 with a 46-10 victory over their Welsh visitors.
And Cockerill said he was delighted for “the most important people at the club” on the pitch and in the stands.
A crowd of more than 19,000 celebrated tries from Tom Croft (2), Martin Castrogiovanni, Ben Youngs, Thomas Waldrom and Matt Smith.
“We spoke during this week about the club being about the players and the supporters, they make the place tick,” said director of rugby Cockerill.
“The support was fantastic and the players were too, from 1 to 23.
“What I wanted, and what the players wanted, was to show the supporters and the rugby world how much they care about what they do.
“The team showed it’s not about the money or playing for their country or anything else, but about playing for Leicester Tigers, playing for the shirt and showing supporters that they care about this club.
“The support was fantastic, it’s a pleasure to play in front of a big crowd and you want to make the best of it and use their enthusiasm too.
“It was probably about time we produced a performance like that for the home crowd.”
With Toby Flood, Anthony Allen and Marcos Ayerza in the starting XV after recovering from injury, there were plenty of reasons to be cheerful.
“It was probably our best all-round game of the season so far,” Cockerill added.
“We want to go on this competition but let’s not get ahead of ourselves, we’ve still got four tough games left in the pool. We let ourselves down in Europe last season and we know we’ve got tough fixtures to come.
“That’s 34 tries now in seven games. We knew we could score tries, but our defence was also outstanding today. The willingness to play without the ball and not let Scarlets have it for long periods was outstanding.
“Now we have to concentrate on Bath next Saturday and back up that performance with another big game."
Tigers host Bath in Aviva Premiership Rugby action at Welford Road on Saturday (2.30pm). Tickets are available now, via the Tigers ticket office. Phone 0844 856 1880, book online or call in at Welford Road (open 8.30am-5pm weekdays, until 8pm on Wednesdays and from 9am on matchday).