The Tigers had 11 players out on international duty and another three injured - which became four just before kick-off when England ace Matthew Tait pulled out of the game with a groin strain.
Despite all that, Leicester were still in with a chance late on but, when 18-year-old fly-half protege George Ford failed with two highly-pressurised drop-goal attempts to win the game, Exeter clung onto a 30-28 lead and recorded a famous victory.
Tigers coach Cockerill said: "We could have snuck it at the end. The battle was good and we just fell short at the end although you have to give credit to Exeter and what they came with.
"I'm not downhearted at all. We have a lot of blokes missing - and I'm not using that as an excuse - but I was pleased with the way we battled away.
"We will review the tape on Monday and we have to improve on some bits, there is no getting away from it.
"The attitude was outstanding, our commitment was outstanding which put us in the mix at the end to potentially win it."
The game was an end-to-end affair from the moment Sireli Naqelevuki gave Exeter the lead from the kick-off.
Niall Morris (two) and Thomas Waldrom crossed in response for Leicester, with Richard Baxter and Matt Jess adding Exeter's other scores.
The rest of the points came from the boot, and Ford, who had a more-than sound game and kicked a first-half drop goal, could have won it had he found his range when two opportunities came his way late on.
Cockerill did not blame the talented teenager, though, saying: "If Fordy dropped the goal at the end, it is a different conversation.
"I don't blame him. He misses those drop-goals under a huge amount of pressure. He's capable of doing it and good enough but on the day he didn't. I don't see that as a negative in his game. That's life.
"George Ford had a good game and, if you get it, you are a hero."
There was nothing other than a big smile on the face of Exeter coach Rob Baxter, though.
It was a momentous victory for the Chiefs, who pushed the Tigers extremely close 12 months ago before losing - a real achievement during their first top-flight campaign.
This time, they clung on and Baxter said: "We are fantastically pleased to have started the season with four points but, at the same time, we are aware that it is a pretty heavily depleted Leicester squad.
"Having said that, it was still very tough. That is two years in a row where we have got away from Leicester on the scoreboard and they very nearly got back to us again today.
"If there is anything I am a little disappointed with it is that we opened up some space on the scoreboard and then made some relatively soft errors.
"But something I am really pleased with is that we didn't crack when the pressure came on. We held it together enough to force a win. So I'm massively pleased but I am still aware that it was not Leicester at full tilt and we still have a lot to work on."
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