Ignacio Mieres scored a last-minute try as Exeter overturned a 23-7 half-time deficit to win 30-23 at Edgeley Park.
Sale had stormed into a seemingly comfortable 20-0 lead after 18 minutes, through tries from Henry Thomas and Sam Tuitupou.
But Exeter never gave up and put on a superb second-half showing to steal the win at the death when Ignacio Mieres grabbed hold of Lavea's stray pass to give them their first Aviva Premiership victory in five games.
However, the first period was very un-Exeter like, with their defence parting like the Red Sea on numerous occasions. And Sale's Nick Macleod opened the scoring with a penalty before adding another after seven minutes.
More joy was to follow for the Sharks when quick thinking from a lineout allowed Rob Miller to set up Thomas. Sale's fly-half added the extras before Dwayne Peel's intelligent kick set-up Tuitupou, who scampered over.
Despite being outplayed, Exeter hit back when a Gonzalo Camacho break and off-load to Richard Baxter reduced the arrears to 20-7.
But Sale continued to threaten the visitors defence and were rewarded when Macleod added another three-pointer before half-time.
However, the Chiefs came out in the second period revitalised. Gareth Steenson reduced the deficit with a penalty before Sharks' Thomas was sin-binned for repeated infringements at the breakdown.
Substitute Mieres converted the resultant attempt and then added the extras when they were awarded a penalty try. Mieres levelled the scores before dramatically stealing victory with a last-minute try.
Tries: Thomas, Tuitpou
Cons: Macleod 2
Pens: Macleod 3
Tries: Baxter, Mieres
Cons: Steenson, Mieres
Pens: Steenson, Mieres 2
Sale: 15 Rob Miller, 14 Mark Cueto, 13 Johnny Leota, 12 Sam Tuitupou (capt), 11 Joaquin Tuculet, 10 Nick Macleod, 9 Dwayne Peel, 8 Andy Powell, 7 David Seymour, 6 James Gaskell, 5 Kearnan Myall, 4 Fraser McKenzie, 3 Henry Thomas, 2 Marc Jones, 1 Alasdair Dickinson.
Replacements: 16 Tommy Taylor, 17 Vadim Cobilas, 18 Tony Buckley, 19 Richie Vernon, 20 Onosa'i Auva'a. 21 Cillian Willis, 22 Tasesa Lavea, 23 Luther Burrell.
Exeter: 15 Luke Arscott, 14 Gonzalo Camacho, 13 Nic Sestaret, 12 Jason Shoemark, 11 Matt Jess, 10 Gareth Steenson, 9 Pat Phibbs, 8 Richard Baxter, 7 James Scaysbrook, 6 Tom Johnson, 5 James Hanks, 4 Tom Hayes (capt), 3 Hoani Tui, 2 Neil Clark, 1 Brett Sturgess.
Replacements: 16 Simon Alcott, 17 Ben Moon, 18 Craig Mitchell, 19 James Phillips, 20 Ben White, 21 Haydn Thomas, 22 Ignacio Mieres, 23 Josh Tatupu.
Referee: Tim Wigglesworth