A stoppage-time penalty by Lee Trundle saw Swansea end Boston's unbeaten home record and consolidate their own position in the automatic promotion places.
It was a dramatic end to a match which had seen fortunes see-saw throughout the 90 minutes.
Swansea were the first to carve out an opening when Trundlle latched on to a Leon Britton pass after five minutes, but home keeper Nathan Abbey dashed out to block.
But it was Boston who snatched a 15th minute lead when Andy Kirk fired home from 12 yards after Lee Beevers had flicked on a David Noble corner. Four minutes later Noble almost extended the home side's lead, but his curling 24-yarder was pushed away by Brian Murphy.
Swansea hit back and exerted spells of pressure, but all they had to show were tame shots from Trundle and Paul Connor which were easlily gathered by Abbey.
The visitors went close to levelling after 33 minutes when a deep Andrew Gurney corner found Connor, whose curling 18-yard drive rattled the bar.
Boston went close again in the 48th minute when a weaving run by Danny Thomas set up Noble, but his 20-yard drive was superbly pushed away by Murphy.
But in the 52nd minute Trundle fired Swansea level when, racing on to a threaded pass by substitute Andy Robinson, he shot beyond Abbey. Three minutes later Trundle again went through, but this time fired straight at Abbey.
Swansea kept up the pressure and went ahead in the 62nd minute when Adrian Forbes crossed for Andy Robinson to stab in.
Boston weren't behind for long and equalised when Simon Rusk headed in at the near post from a Thomas cross.
Boston pressed again when Kirk latched on to a Jason Lee chest down from a Thomas pass, but he fired tamely at Murphy.
After Trundle had an effort scrambled off the home line by Beevers, Swansea won it when the same player was tripped by Lee in stoppage time and picked himself up to fire home a penalty.