Leicester had not lost at home in the Premiership to anyone other than Saracens since September 2008
Exeter recorded a famous win over weakened Leicester in a pulsating Premiership encounter at Welford Road.
Sireli Naqelevuki went in for the Chiefs in the opening minute before Thomas Waldrom answered with a try for the hosts two minutes later.
Niall Morris put Tigers ahead but Tom Johnson's score and eight points from Ignacio Mieres kept Exeter ahead.
Matt Jess crossed after the break, before a second try for Morris, but the visitors clung on for victory.
The Chiefs have now beaten the Tigers twice this year, following their LV= Cup win in January.
But Leicester, missing 11 players on World Cup duty, were uncharacteristically error-strewn in defence right from the off as they suffered an opening-day defeat for the third season running.
It was the worst possible start for the hosts as Naqelevuki's try and the conversion by the impressive Ignacio Mieres left them a score down.
But Waldrom's reply, converted by Billy Twelvetrees, was followed by a drop-goal by Tigers' teenage fly-half George Ford.
Mieres hit a penalty before Morris, in as a late replacement for the injured Mathew Tait to play his first game for Leicester since his summer move from Heineken Cup holders Leinster, went in for a debut try.
The hosts could not convert but, five minutes later, Mieres was again on target, after Johnson touched down, and the Argentine put the Devon men ahead with two more penalties up to half-time.
Twelvetrees made it 23-18 in the second half before more defensive errors allowed Jess to score and, with the conversion, Leicester were suddenly 12 points down.
Tigers added a penalty and Morris got his second try but the introduction of Gareth Steenson off the Exeter bench set up a dramatic finish.
The replacement stand-off had three golden chances to put Exeter out of sight but fluffed the trio of place kicks.
The home side rallied and twice Ford went for the winning drop-goal in the closing minutes but the England Under-20s stand-off pushed both attempts wide.
Leicester coach Richard Cockerill: "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.
"The attitude was outstanding, our commitment was outstanding which put us in the mix at the end to potentially win it."
Exeter coach Rob Baxter: "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.
"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.
"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."
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