The outcome left the Devon side just one point behind the Tigers, who saw their run of five league wins halted.
Head coach Rob Baxter named virtually a full-strength Exeter side having rested many for the recent LV= Cup ties as the Chiefs faced the first of three games out of four at Sandy Park.
Fly-half Toby Flood was in the Leicester starting line-up after a knee injury, while centre Manu Tuilagi was included for the first time since a hamstring problem in December.
The Chiefs knew that they had to start well and pinned the Tigers back in their own half for the opening minutes - but unforced errors crept into the play of both sides and attacks petered out.
The stalemate was broken after 10 minutes as Chiefs full-back Phil Dollman was unable to stop Samoan wing Alesana Tuilagi powering through to touch down in the corner. Flood was off target with the touchline conversion attempt.
Exeter had problems at their own lineouts while they questioned some decisions of referee Sean Davey, who was reported by Baxter earlier in the season following their 19-13 defeat at Harlequins when he was unhappy with Davey's handling of the scrum and breakdown.
Former England prop Julian White later received a stern lecture from the Sussex match official, who had looked to be going to his pocket to pull out a yellow card.
Approaching the half-hour mark, Flood kicked a 20-metre penalty after Chiefs skipper Tom Hayes was penalised for not rolling away at the tackle.
But Exeter hit back immediately from the restart as Argentinian fly-half Ignacio Mieres produced an angled kick from just outside the Tigers 22 which sparked real life into the home side.
Mieres saw an opportunity looming out on the right and his chip to the corner was taken by wing Gonzalo Camacho but he was tackled by Manu Tuilagi.
Exeter maintained the pressure and were denied another scoring opportunity when scrum-half Sam Harrison was forced to touch down over his own line. Leicester were again penalised but Mieres pushed his attempt past the far upright.
On the stroke of half-time Mieres had another opportunity to reduce the arrears with a 35-metre penalty after the Tigers were caught offside, leaving the Chiefs trailing 8-6 on a a bitterly cold evening.
Leicester looked the stronger side at the start of the second half and when Manu Tuilagi burst through to the Chiefs half the home side were penalised for going off their feet - but Flood was short with his 45-metre attempt, although soon after he slotted his second kick through the uprights from 15 metres out.
Exeter heads did not drop though as they hit back immediately. England Saxons flanker Tom Johnson charged down a kick and number eight Richard Baxter raced through to touch down, with Mieres' touchline conversion nudging the Chiefs in front.
White was again penalised for offside at the breakdown and Mieres' 25-metre penalty attempt hit the far upright and bounced over the bar, followed by another from a similar position - giving the Chiefs an eight-point advantage with 12 minutes remaining.
Flood then saw his 35-metre penalty attempt hit the upright and bounce back into play, much to the relief of the majority of the 9,025 crowd.
Play was held up for three minutes when Chiefs prop Brett Sturgess received treatment for an injury to his right shoulder.
Leicester knew they had to score twice as the final six minutes ticked down but the Chiefs defence stood firm with some solid tackling and the home side went closest to scoring in the closing stages through centre Nic Sestaret, who was stopped just short.
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