The game turned after 10 minutes as Sarries prop Richard Barrington was sent off for a high tackle.
Gareth Steenson converted the resulting penalty and Jack Nowell's try gave Chiefs a 10-3 lead at half-time.
Saracens performed admirably after the break and rescued two points when Titi Lamositele was forced over the line and Alex Lozowski converted.
Sarries, who can be usurped at the top if Wasps avoid defeat against Leicester on Sunday, lost Marcelo Bosch with a head injury after just five minutes, with Steenson missing a penalty four minutes later.
But the game's controversial moment was just seconds away as Brad Barritt initially tackled Geoff Parling high, with Barrington following up with his shoulder into the head of Exeter's former England lock.
Under the new interpretation of the high tackle rules Barrington was sent off, with Barritt lucky to avoid any sanction, although he may well be cited after the game.
Exeter took advantage almost straight away as Nowell raced into the corner after an impressive one-handed catch from Steenson's perfectly weighted chip.
Lozowski - replacing the rested England fly-half Owen Farrell -  pulled three points back with a penalty, but Exeter were almost further ahead after Don Armand broke through the line on a 30-metre run, only to be excellently stopped by Chris Wyles and Michael Rhodes.
After the break, Saracens upped the tempo and did not look like a team with a one-man disadvantage - Lozowski converting a penalty as Steenson missed a similar chance to restore the lead.
Exeter's set-piece failed to function as Saracens piled on the pressure, twice going close to the Chiefs try-line before Steenson got his second penalty with 11 minutes to go.
But more Saracens pressure finally told as Lamositele was forced over late on and Lozowski added the extras to secure what, at the start of the game, seemed an improbable draw.
Saracens director of rugby Mark McCall:
"The players were incredible with how they coped and dealt with all the situations that arose, and by the end it was a game that we maybe could have won as well.
"I felt Geoff was going to the ground and Richard (Barrington) came in as the adjusting defender, and he made contact. Obviously we hope Geoff's okay.
"Today ended up being a titanic struggle between 15 and 14, but I don't think anyone wants to see those kinds of games.
"I think we need to be sensible, I'm not saying they weren't today, but we've got to be sensible with how we apply this care for players, and that's all."
Exeter head coach Rob Baxter told BBC Radio Devon:
"The players are a little annoyed, and so they should be. Especially in the second half we've got to have a little bit of a look at ourselves and say why did we not really feel like we took the game by the scruff of the neck? Why were 14 men more disciplined than us? Why did 14 men drag that game their way?
"You've got to give Saracens a great deal of credit and in some ways I'm not sure who the red card affected the most, as in the second half it seemed to stop us playing a little bit.
"Great credit to Saracens, they showed a lot of the qualities that you want to talk about when you talk about Saracens.
"It's good if we're a little bit upset and we're burning a little bit and we want more out of games like that."
Saracens: Goode; Maitland, Bosch, Barritt (capt), Wyles; Lozowski, Wigglesworth; Barrington, Brits, Figallo, Itoje, Hamilton, Rhodes, Brown, Wray.
Replacements: George, Lamositele, Koch, Skelton, Burger, Spencer, Taylor, Tompkins.
Exeter: Dollman; Nowell, Hill, Slade, Short; Steenson (capt), Chudley; Rimmer, Cowan-Dickie, Holmes, Atkins, Parling, Armand, Johnson, Waldrom.
Replacements: Yeandle, Moon, Williams, Lees, Dennis, Maunder, Whitten, Campagnaro.
For the latest rugby union news follow @bbcrugbyunion on Twitter.
Fourteen-man Saracens battled to an impressive draw with Exeter to go top of the Premiership.