Despite being outclassed by City for much of the game, Spurs had twice come from behind as goals from Erik Lamela and Lucas Moura - 19 seconds after his introduction - cancelled out efforts from Raheem Sterling and Sergio Aguero.
Jesus thought that he had earned City three deserved points when firing home in the second minute of added-on time, only for VAR to rule that Aymeric Laporte had handled the ball in the build-up.
The decision leaves each of City and Spurs two points off the pace after Arsenal and Liverpool got the better of Burnley and Southampton earlier on Saturday.
City wasted no time in penning Spurs back into their own defensive third during the early stages, although the home side's only effort of any note was a goalbound strike from Sterling which was well blocked by Kyle Walker-Peters.
The reigning champions could have been handed a spot kick soon afterwards when Lamela hauled down Rodri from a corner, but neither the referee nor VAR deemed that an offence had been committed.
However, it was not long before City had their first goal. A superb deep cross from Kevin De Bruyne found Sterling on his own at the back post, and the winger kept his composure to send a header into the far corner of the net.
There were signs that City could quickly double their advantage but three minutes later, completely against the run of play, Spurs found an equaliser as Lamela curled a low effort into the bottom corner from 20 yards.
Rather than knock City out of their rhythm, there was an increased sense of urgency from Pep Guardiola's side, and it came as no surprise when they regained the lead 10 minutes before the break.
De Bruyne was again the creator - this time from the byline - and Aguero had the relatively simple task of converting from inside the six-yard box after getting in front of his marker.
City picked up where they had left off once the game had restarted, with Oleksandr Zinchenko having a long-range strike well kept out by Lloris, and both De Bruyne and Aguero firing off target from promising positions.
City required no invitation to go straight back onto the offensive as Rodri saw a fizzing drive tipped over the crossbar by Lloris, before Nicolas Otamendi had a header hooked away inside the six-yard box.
However, City's growing momentum was halted midway through the second half as Guardiola substituted Aguero, which resulted in a heated confrontation between the pair as the striker expressed his unhappiness over being withdrawn.
Jesus had a couple of opportunities after his introduction, seeing a deflected shot go over the crossbar and a tame header bouncing just wide of the far post.
While City remained in the ascendancy heading into five minutes of added-on time, Spurs were doing enough to keep openings down to a minimum as they moved closer to a hard-earned point.
That effort appeared to be for nothing when Jesus kept his composure to fire into the far corner from 10 yards only for VAR to have the last say, despite no appeals from Spurs that Laporte had made any connection with the ball using his hand.
Nevertheless, like Wolverhampton Wanderers encountered against Leicester City last weekend, the rule regarding handball was applied to the letter of the law, although City will feel hard done by on a day where Arsenal and Liverpool have gained the initiative at the top of the table.
MANCHESTER CITY (4-2-3-1) Ederson; Walker, Otamendi, Laporte, Zinchenko; Rodri (D.Silva 78'), Gundogan; Bernardo (Mahrez 80'), De Bruyne, Sterling; Aguero (Jesus 66')
TOTTENHAM HOTSPUR (4-2-3-1): Lloris; Walker-Peters, Sanchez, Alderweireld, Rose; Winks (Moura 56'), Ndombele; Sissoko, Eriksen (Skipp 90+1', Lamela (Lo Celso 85'); Kane