Davis added the conversion and the Chiefs were looking comfortable as they led 10-3.
Sale kept in touch with two further penalties from Macleod, the first a huge kick from 55 metres out that was helped by the strong breeze.
On the stroke of half-time Davis had the chance to extend the lead once again, but his penalty bounced back off the uprights and the half ended with Exeter leading 10-9.
There was no let up in intensity at the start of the second period but the hosts now had the Sandy Park wind behind them and it did not take long for Rob Baxter's men to trouble the scoreboard again.
Flanker Johnson was driven over in the corner after 43 minutes as Exeter's pack again surged forward from an attacking line-out.
However, any hopes of breathing space were short-lived, as Sale responded almost immediately with a try from Crane, the winger cleverly put through to score by England centre Matthew Tait.
Macleod was on target with the conversion, cutting Exeter's lead to just one point once again.
With a barely a moment to pause for breath, the 51st minute saw former Bath winger Higgins cross the whitewash for his first ever try in an Exeter shirt, soothing the frayed nerves of the vociferous home support, as Davis added the crucial conversion.
In a thrilling climax Sale pushed desperately for the win, a late Macleod penalty reducing the deficit to just five points, but the Devon outfit kept their composure to record a fifth Premiership win from nine games.
Exeter Chiefs backs coach Ali Hepher told BBC Radio Devon: "We grafted real hard for that, we could have made it a little bit easier for ourselves, but all in all we're still delighted.
"We haven't really looked at the long-term, we've got short-term goals to work on, which to be fair the guys have achieved today.
"At the end of the day the league table will sort itself out and we've just got to keep concentrating on making the right decisions and getting wins."
Sale head coach Mike Brewer said: "We came here with a certain style that we wanted to play which was based on set-piece but the set-piece in general, but the line-out in particular, was dismal.
"That was the reason we couldn't execute what we set out to do and when we did get a little bit of ball in the second half we scored a try and again towards the end of the game when we stretched the play across the field.
"In the first half we turned the ball over too soon, our kick-off receipt was poor, guys getting in the wrong position resulted in really poor performance in the first half and let Exeter in to the game.
"When we scored points we gave the ball straight back to them and we knew that they would target us through the middle where we were a little soft earlier in the season."
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