Sean Dyche's men withstood the pressure in front of a sparse crowd at Wembley, thanks partly to two excellent saves from goalkeeper Joe Hart, until substitute Eriksen scored in the 91st minute.
Mauricio Pochettino made five changes to the team that drew with Barcelona this week, handing a first Premier League start to 18-year-old Oliver Skipp, and his side certainly lacked their customary cutting edge against a rigid Burnley rearguard.
However, the introduction of Eriksen proved crucial as Spurs moved back to within five points of league leaders Manchester City.
Lucas Moura, who scored the crucial equaliser against Barca, was among those restored to the starting line-up and almost broke the deadlock with an improvised effort from Erik Lamela's long ball.
Harry Kane had shouts for a penalty waved away but it was 30 minutes in until the first effort on target, which saw Lamela denied by some good work from Hart.
Burnley had barely threatened early on but could twice have gone ahead, with Ashley Barnes heading over after an error from Hugo Lloris and then seeing a goalbound volley blocked.
The visitors' time-wasting tactics were beginning to irritate the Spurs fans and players, but when Lamela finally had another clear opening 16 minutes from time, Hart again responded with a fine one-handed save.
Substitute Son Heung-min scuffed a shot wide and Dele Alli failed to convert his cross with the goal gaping, but Eriksen eventually proved the hero, side-footing home Kane's pass to break Burnley hearts.
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As it turned out, they looked lethargic after their Champions League heroics, even amid the changes in personnel, and offered very little consistent threat to Hart's goal despite having 71 per cent of the possession.
Hart heroics deserved better
Tottenham only managed three efforts on target: two of them were saved expertly by Hart, and the third was a goal he could do little about. The England goalkeeper did not deserve to be on the losing side.
Frustrated Kane kept subdued
Kane took exception to some of Burnley's delaying tactics, and his frustrations were reflected in a poor display at the spearhead of the attack, even if he did manage an assist.
Key Opta Facts
- None of Tottenham's last 21 Premier League games have ended level, with Spurs winning 16 and losing five since a 1-1 draw at Brighton in April.
- Only Bolton Wanderers, in 2011-12 (18), have had a longer run without a draw from the start of a Premier League campaign than Spurs this season (17).
- Burnley failed to register a single shot on target in this match. It’s the third time they've done this in the Premier League this term – no other side has done so more than once.
Spurs now turn their attention to a derby trip to Arsenal in the EFL Cup on Wednesday, before they resume league duties away at Everton. Burnley have a week off ahead of their own visit to the Gunners.