Thompstone, signed from London Irish in the summer, scored one in the first half and two in the second, with England fly-half Toby Flood kicking 15 points in a fine all-round performance. Replacement Ben White scored a last-minute consolation try for Exeter.
Despite suffering from an injury crisis, Leicester were too strong for an Exeter side who showed none of the form which earned them the double over the Tigers last season and the scalp of Saracens last week.
Leicester's injury problems intensified when three players - hooker George Chuter, full-back Scott Hamilton and prop Boris Stankovich - withdrew on the day of the game, taking the total on the treatment table to 12.
They were also under pressure after last week's home defeat by leaders Harlequins and having gone 166 minutes without scoring a try, their worst barren run since November 2010.
In contrast, Exeter made just two changes, England international Tom Johnson coming into the back row and Kiwi Jason Shoemark into the centre.
Following their success here last term, Exeter were attempting to join a select band of teams - Northampton, Saracens, Harlequins and Wasps - who had managed back-to-back league wins at Welford Road.
And they drew first blood when fly-half Ignacio Mieres banged over a 49-metre penalty after just two minutes.
Leicester, however, hit back in impressive style and dominated the rest of the half with the pack taking control and Flood pulling the strings behind.
It was Flood's inside pass to Vereniki Goneva which gave Leicester the breakthrough, the Fijian winger bursting into the Exeter 22 and Thompstone finishing off the move with a try in the right corner.
Flood kicked the touchline conversion and made in 10-3 six minutes later with a penalty from a similar position.
Mieres missed with a penalty from a metre inside his own half before Johnson, prop Brett Sturgess and centre Sireli Naqelevuki sparked Exeter's best move which only failed to produce a try because of some great scrambling defence by Leicester.
Leicester turned defence into attack after a great scrum on their line and then rejected the chance of another three points by opting for the line out and drive, the move breaking down when Flood failed to find Goneva with a long floated pass.
Flood did, however, end the half on a high note with penalties in the 30th and 40th minutes to stretch Leicester's lead to 16-3.
Exeter started the second half strongly, Naqelevuki bursting into the 22 before ruining his good work with an attempted pass behind his back which went forward.
And Leicester made the visitors pay by working their way up field and scoring their second try through Thompstone, his former Irish team-mate Dan Bowden providing the
scoring pass with Flood adding the conversion.
The try was made by the pressure Thomas Waldrom put on the Exeter defence and also the fact that he took out Mieres off the ball, an incident the referee missed and the TMO did not appear to take into account.
With their tails up the Tigers piled on the pressure and giant Samoan prop Logovi'i
Mulipola had an easy chance to score but lost the ball in the tackle over the line.
Exeter, however, began to find some form and exerted strong pressure on Leicester without rattling the scoreboard. After a couple of good moves down either wing they went for the catch and drive but Leicester stole their line out ball.
And Leicester stretched their lead to 30-3 with 13 minutes left after a turnover in their own half. Flood countered and Matt Smith chipped through for Thompstone to complete his hat-trick. Flood added the conversion.
With the last attack of the game, replacement White scored for Exeter from close range, the TMO awarding the try after long deliberation, but Gareth Steenson missed the conversion.
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