In-form striker Emmanuel Adebayor scored either side of Chico Flores' own goal, with Wilfried Bony netting Swansea's consolation, as Tim Sherwood's men moved level with fourth-placed Liverpool, extending the hosts' winless league run to eight matches.
The last time Spurs won five league matches in a row on the road was in the 1960-61 campaign, when they completed the top-flight and FA Cup double.
Despite being on the back foot for much of the first half, the visitors went in front 11 minutes before the break thanks to Adebayor's header from Christian Eriksen's superb delivery.
Chico was the man at fault for Tottenham's second as his touch took Kyle Walker's cross past Gerhard Tremmel to put the game out of Swansea's reach eight minutes after the break.
There was time for Adebayor to complete his brace with a cool finish from inside the area and, despite Bony's strike, Sherwood maintained his unbeaten league record in charge of Spurs.
He had Swansea's first effort on target after 13 minutes, shrugging off the attentions of Vlad Chiriches before forcing Hugo Lloris into a smart stop with a long-range drive.
Spurs' attack had looked out of sorts for much of the half but Adebayor found the net with their first effort of the game.
Eriksen's pinpoint cross was met by the former Arsenal striker, who registered his fifth goal in eight games from close range.
An own goal from Chico double Spurs' advantage early in the second half, with the Spaniard diverting Walker's cross into his own net after Roland Lamah had failed to track the right-back.
Adebayor scored his second 19 minutes from time, calmly rolling Danny Rose's ball past Tremmel, before Bony eventually got on the scoresheet with an expert volley after 78 minutes.
His 14th goal of the season made little difference to the scoreline, though, with Laudrup's men left three points above the relegation zone ahead of clashes with Fulham, West Ham and Cardiff City - all of whom occupy spots in the bottom four.
