Mauricio Pochettino's men were frustrated for the much of the match as Burnley were on course for only their second away point of the season, but Rose's stunning effort 19 minutes from the end moved Spurs to within a point of fourth-placed Arsenal after their 2-1 loss in Manchester.
Although Spurs exerted control and saw several opportunities go to waste early in the first half, Burnley broke the deadlock thanks to Ashley Barnes – only their second Premier League goal on their travels this season.
The hosts grabbed a swift equaliser thanks to Dele Alli, though that failed to inspire a great improvement as Burnley's stubborn defence held firm for the rest of the half.
But his efforts were ultimately for nothing, as Rose's rasping drive found the net and wrapped up the three points in Spurs' final match before Christmas.
Pochettino's side made a rapid start as Rose's fine pass up the left flank released Alli in the area, only for the midfielder to slice over from a good position in the second minute.
But the hosts were lucky to remain level shortly after. Kyle Walker was robbed by Stephen Ward as he tried to play himself out of danger, and Andre Gray slammed the Irishman's eventual cross at Hugo Lloris from eight yards.
Burnley were not to be denied in the 21st minute, though. Spurs failed to clear an aerial delivery and George Boyd scuffed the ball back into the danger zone for Barnes to stab home from close range.
But their lead lasted just six minutes, Alli arriving in the area late and meeting Walker's fierce low cross with a clinical finish into the bottom-right corner.
Harry Kane almost completed the turnaround just before the break when he raced on to Eriksen's throughball and fired at Heaton.
Burnley were furious when Eric Dier tackled Gray from behind in the penalty area and referee Kevin Friend refused to point to the spot eight minutes after the restart.
Spurs then went straight down the other end and should have taken the lead.
Rose charged down Matt Lowton's attempted clearance by the corner flag and then volleyed a cross into the goalmouth which Eriksen headed straight at Heaton with little conviction.
The Denmark international was a central figure again just after the hour-mark, with his dipping 20-yard drive forcing a strong save from Heaton, but the goalkeeper was eventually beaten.
The much-maligned Moussa Sissoko, who replaced Mousa Dembele in the 63rd minute, did brilliantly, charging through the centre and then playing a pinpoint pass into the left side of the area for Rose, who struck a stinging drive into the top-left corner.
Heaton was called into action again to deny Son Heung-min from a tight angle in the closing stages, but Burnley never looked punishing them at the other end for a fifth loss in six matches.