A stellar performance from Hugo Lloris kept the Foxes largely at bay with the pressure now really mounting on Nigel Pearson following yet another defeat.
Christian Eriksen's 70th-minute free-kick secured all three points for Mauricio Pochettino's side, after they had soaked up pressure from their hosts.
Harry Kane gave Tottenham a superb start with his third league goal in as many games in the second minute, but the visitors were unable to pull further clear prior to the interval.
Rock-bottom Leicester, who had manager Nigel Pearson serving a one-match touchline ban for abusing a fan, levelled three minutes after the break through Leonardo Ulloa.
Chances continued to come for Pearson's men thereafter, but they were punished for their wastefulness when Eriksen caught Ben Hamer off-guard from a set-piece.
Tottenham have now won three league games in succession, while Leicester remain rooted to the foot of the table despite an improved display.
Having failed to win any of their last 12 league matches, Leicester would have hoped for a solid start, yet they fell behind almost immediately.
Erik Lamela released Nacer Chadli in behind the Leicester defence and the Belgium international rolled a low cross to Kane, who touched the ball away from Wes Morgan before firing past Hamer.
Ulloa – one of two changes for Leicester – struggled to get involved in the opening exchanges, but brought a save out of Hugo Lloris after 18 minutes.
The Argentine striker flicked up a pass from Jeffrey Schlupp and volleyed towards goal from 20 yards, although Tottenham's goalkeeper had no problems tipping it over.
Tottenham, who recalled Danny Rose and Benjamin Stambouli, were posing much more of an attacking threat than the hosts and were denied a second goal on two occasions by last-ditch tackles.
On the half hour, Marcin Wasilewski prevented Lamela from converting a tap-in after neat work from Kane and Christian Eriksen, before Paul Konchesky got back just in time to halt Nabil Bentaleb three minutes later.
Leicester should have gone into the break level but Lloris saved well from Riyad Mahrez and, even when the Frenchman was beaten, the post kept out Ulloa's header.
Ulloa was not to be denied early in the second half, though, as a cute flick converted Mahrez's low cross from the right.
Leicester should have been ahead four minutes later, but Mahrez's free-kick rattled the frame of the goal from 20 yards, with Lloris a mere spectator.
Leicester were now in the ascendancy and it took a superb save from Lloris to keep the scores level. Schlupp skipped past Kyle Walker but his left-foot drive was tipped wide by the France international.
Eriksen made the home team pay for their profligacy with 20 minutes remaining as he caught Hamer off-guard with a long-range free-kick, the Denmark midfielder beating the goalkeeper at his near post.
Leicester pushed for an equaliser with the introduction of Jamie Vardy and Anthony Knockaert, but it was
Morgan who went closest to levelling.
However, Lloris was equal to two efforts from the Leicester captain to secure the victory and extend City's winless run in the league to 13 matches.