Sergio Aguero was also back on the bench after his thigh problem although Jesus proved the perfect deputy with a sharp left-footed finish in the 22nd minute and a 50th-minute header to register his first Premier League goals since August.
City manager Pep Guardiola was delighted with the Brazilian's performance.
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"He showed fight and personality, he is a young player and has to improve many things. It is good for his confidence," said the Spaniard.
Guardiola, whose team has 44 points from 17 games while Liverpool have 42 from 16, added that he was looking forward to tuning in to Sunday's clash at Anfield. "Yes it is a nice game to watch," he said.
At Wembley, a Burnley side inspired by goalkeeper Joe Hart thought it had secured a point with an impressively resilient performance that frustrated Spurs.
Yet Eriksen, who came off the bench in the 65th minute, finally found a way past the former England keeper in the dying seconds to secure the three points.
"Every time you come on you want to make an impact. To score in extra-time was extra nice," said Eriksen.
Wolves beat Bournemouth 2-0 with goals from Raul Jimenez and Ivan Cavaleiro to win a third successive top-flight game for the first time since 1980 and move into seventh place.
Wolves manager Nuno Espirito Santo was delighted by the way his side braved the elements on a bitterly cold day across England.
"In this weather, with the wind, cold and rain and against Bournemouth - I'm very proud. We did a fantastic job," he said.
The two sides had never met at this level before but the result was seldom in doubt and Bournemouth continue to drop down the table after losing a sixth game out of seven.
Below them, Cardiff City remain in the thick of the relegation fight after losing 3-2 at Watford, a game graced by a stunning opener from the home side's Gerard Deulofeu, who slipped past two defenders in the box before drilling a fine finish past goalkeeper Neil Etheridge.
Watford's Portuguese youngster Domingos Quina, 19, scored his first Premier League goal before Cardiff fought back with two goals in the last 10 minutes, but they were unable to stop the home side winning their first league match in seven.
Crystal Palace finally broke their curse of not being able to win without talismanic winger Wilfried Zaha in the side as they beat Leicester City 1-0.
The Londoners, who last won without the suspended Zaha in September 2016, had Luka Milivojevic to thank for a win that moves them clear of the relegation zone.
Newcastle also boosted their survival chances with an opportunistic 1-0 victory at strugglers Huddersfield Town. In-form striker Salomon Rondon scored the only goal from a visiting counter-attack in the second half.
"Against a team so close to us in the table, and to get three points, it was a good achievement," said Newcastle manager Rafa Benitez.
In the day's late game, Fulham suffered a 2-0 defeat to West Ham United that leaves the Cottagers stuck to the bottom of the table and elevated the Hammers into the top half of the table.
Reuters