But in-form Burnley posed a strong challenge and the Clarets took the lead through Chris Wood's header from a Dwight McNeil corner the visitors felt was incorrectly awarded.
Kane, who was sidelined for six weeks by an ankle problem, appeared to have claimed a point for Spurs but Barnes - who has now scored in four consecutive top-flight outings - turned in decisively from close range with seven minutes to go.
4 – Ashley Barnes has scored in each of his last four Premier League games (4 goals), after netting in just three of his previous 24 in the competition (4 goals). Bundle. #BURTOT pic.twitter.com/BgIUY5G3Yq— OptaJoe (@OptaJoe) February 23, 2019
Hugo Lloris looked uncomfortable under a couple of early high balls, while the returning Kane rifled a left-footed drive wide in the 16th minute.
As the game opened up, Tom Heaton easily denied Christian Eriksen before Barnes whistled a first-time strike narrowly over the crossbar.
Kane's swerving 25-yard strike forced a superb save from Heaton after the break but it was the Clarets who took the lead in the 57th minute.
McNeil's in-swinging corner from the right wing found Wood rising above the Tottenham defence to nod home his sixth goal in eight Premier League matches.
Mauricio Pochettino sent on Fernando Llorente to join Kane in attack and the England captain levelled just eight minutes after Wood's opener.
Rose caught Burnley's defence cold with a clever throw-in that sent Kane through to beat Heaton with a cute finish.
But as Spurs pushed for a winner they were downed by Barnes, who showed a poacher's instinct to arrive at the back post and convert a cross-shot from substitute Johann Gudmundsson.
What does it mean? Spurs' title charge halted
A three-way challenge for the league looked on the cards, with Spurs putting pressure on City and Liverpool thanks to a fine run of results despite the absence of Kane in recent weeks.
But defeat at Burnley represents a missed opportunity, especially with City not in league action this weekend - a fact apparently not lost on Pochettino as he raged at referee Mike Dean following the final whistle.
Heaton shines in front of Southgate
Burnley's run of eight league matches without defeat has coincided with the return of club captain Heaton in goal, replacing Joe Hart. He made a wonderful save to deny Kane, which would surely have impressed the watching England manager Gareth Southgate.
Eriksen goes missing
Burnley fell to a late Eriksen goal at Wembley earlier in the season, but the Denmark international offered little at Turf Moor. Kane's return should have given the playmaker more to work with but his impact was minimal.
Key Opta facts
- Burnley are unbeaten in eight Premier League games (W5 D3), their longest top-flight run without defeat since a run of nine in 1966.
- During their current streak, the Clarets have collected 18 points after picking up 14 from their previous 24.
- Wood has netted four goals in his past four Premier League appearances, as many as in his previous 27 in the competition. Barnes had three in 24 before his ongoing run of four in four.
- Kane has netted five goals in his seven Premier League appearances against Burnley, including four in his past three.
- Gudmundsson has registered 15 assists for Burnley in the Premier League, nine more than any other player.