Despite lying 10th in the league, Leicester's victory over Gloucester last weekend left them only four points adrift of the title play-off zone.
And Cockerill is delighted after the start his team have made to a demanding period of games that continues against Saints and continues later this month through home and away Heineken Cup appointments with French heavyweights Clermont Auvergne.
"When Leicester sides come under pressure, they generally come up with the goods. We have really fronted-up these last two weeks," he said.
"This is a tough five-week spell for us, but we are two from two (after victories over Ulster and Gloucester), and the last two weeks have been a truer reflection of where this club is at.
"Hopefully, we are starting to turn the corner, and there are some good players to come back in as well.
"It is always difficult to win at Kingsholm and, in the circumstances of where we are in the table it was an exceptional result."
Gloucester, meanwhile, are back in action on Friday night, making the long trip to bottom club Newcastle, with Falcons currently nine points adrift of 11th-placed Worcester.
Gloucester have not won at Kingston Park since 2007, and their recent form - one victory from six games in all competitions - suggests that wait could continue.
"We probably go in as underdogs, but I am happy with that," Gloucester head coach Bryan Redpath said. "If they are favourites because of their track record against us then great.
"Newcastle away is a massive game for us. We can't rely on our home form at the moment, so we have to rely on some of our away results."
Redpath has made five changes after the Leicester loss, including a return for Samoa centre Eliota Fuimaono-Sapolu, who is back after serving a three-week ban for his latest indiscretions on Twitter.
James Simpson-Daniel, Tom Voyce, Dave Lewis and Scott Lawson make up the remainder of the quintet restored to the starting XV, with Redpath looking for his side to start turning narrow defeats into victories.
Falcons boss Alan Tait, meanwhile, has made seven changes, including call-ups for ex-Gloucester number nine Jordi Pasqualin and Scotland international prop Euan Murray.
Elsewhere in the Premiership on Saturday, Bath host Sale Sharks and Worcester travel to Exeter with Warriors head coach Richard Hill having lost back-row trio Neil Best, Kai Horstmann and Adam Balding through injury.
And it has prompted Hill to enter the player market, with a new arrival at Sixways expected soon.
"We need to bring someone in, and we have just about done a deal with a player who is not from Great Britain," Hill said.
"He will significantly bolster our back row. He will have to go straight in, and we plan to see him before Christmas and for the game with Leicester (on December 27)."
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