Former England number 10 Burns scored 16 points, including an interception try, after coming on as a second-half substitute to see Leicester home against last season’s beaten Champions Cup finalists.
“We know Freddie has got some stardust about him,” rugby director Cockerill said. “You take the risk with the intercept, and if it comes off, you are okay.
“Freddie is a very good footballer and he read it very well. Everything went right for him tonight, and I am delighted for him.
“Freddie is up for the challenge. If he is in the limelight, he is happy!
“Those sorts of personalities play in those sorts of positions, and he would have relished having the opportunity to show what he could do.”
The Tigers knew they had to roar following a 42-13 drubbing by Pool One rivals Glasgow nine days ago — or face having their quarter-final hopes hanging by a thread.
And even though it was far from pretty, they delivered after flanker Brendon O’Connor scored an early try, before two Owen Williams penalties and then Burns’ points haul saw them home.
New Zealand superstar Dan Carter scored 12 of Racing’s points through a try, two conversions and a penalty, while wing Juan Imhoff touched down five minutes from time, yet the reigning Top 14 champions never got going, as Leicester made it 15 games unbeaten against French opponents at Welford Road since Biarritz beat them in January 2005.
The Tigers have still got it all to do in a group that also includes Munster, and their December back-to-back meetings with the province could prove pivotal.
Cockerill added: “I am delighted with the performance and the result, and to deny them anything out of the game is important. We defended well, and our set-piece was probably the difference.
“If we had lost this evening, it would have been very difficult to get out of the pool, especially after last week’s performance.
“We are all aware of the expectation here. We could have lost tonight, but we were in the contest and put in a performance.
“They (Racing) have an exceptional, world-class backline, and if you give them opportunities, they are going to cause you problems, but we controlled them reasonably well.”
Racing had their moments in attack, but were flustered by a combination of poor handling and aggressive Leicester defence, and the Tigers punished them with two Williams penalties during the second quarter, although not before Australia international centre Matt Toomua departed injured.
The game’s scrappy nature did not worry Leicester, with the result far outweighing performance after their capitulation at Glasgow nine days ago, but they allowed Racing back in by conceding a soft try just nine minutes after the restart.
Carter scored it, courtesy of throwing two dummy passes and executing a hand-off on Youngs, yet Leicester were guilty of wafer-thin defending as they allowed the World Cup winner an easy run.
Carter’s conversion cut the deficit to a point, but then it became the Burns show as he kicked two penalties and converted his try for 13 points in six minutes, and Racing were finished, despite a late Imhoff touchdown that Carter converted, before Burns’ 77th-minute penalty sealed the deal.
It was not all trouble-free for Leicester though, and there was a long delay near the end as full-back Telusa Veainu went down injured before being carried off.
Tigers’ Australia international centre Matt Toomua, meanwhile, departed in the first half after hurting his knee.
Leicester: Tries: O’Connor, Burns. Cons: Burns. Pens: O. Williams 2, Burns 3.
Carter, Imhoff Cons: Carter 2 Pen: Carter
Veainu, Thompstone, Tait, Toomua, Pietersen, O. Williams, B. Youngs, Ayerza, T. Youngs, Cole, Slater, Barrow, Fitzgerald, O’Connor, McCaffrey.
Harrison for Veainu (71), Betham for Toomua (23), Burns for O. Williams (54), Mulipola for Ayerza (59), McGuigan for T. Youngs (76), Kitchener for Barrow (18).
Dulin, Rokocoko, Goosen, Chavancy, Imhoff, Carter, Machenaud, Ben Arous, Chat, Tameifuna, Nakarawa, Carizza, Lauret, Nyanga, Masoe.
Afatia for Ben Arous (55), Lacombe for Chat (55), Ducalcon for Tameifuna (54), Van Der Merwe for Carizza (63), Dubarry for Masoe (78).
Marius Mitrea (Italy).