The frontman, on loan from Chester, hit the mark 18 minutes from time to edge the Pilgrims away from the basement zone and leave Accrington seriously in danger of a quick drop back to the Conference.
Stanley arguably enjoyed the better of the early exchanges and created two decent chances.
But first a seventh minute 30-yard free-kick by Michael Welch was well held by home keeper Andy Marriott and then Shaun Whalley fired wildly over the top when an opening beckoned down the left.
Boston hit back with a few waves of pressure culminating in a couple of corners but neither unduly worried the visiting rearguard.
Accrington went close again in the 21st minute when a low corner by Andrew Proctor was met by Andy Todd, whose angled snapshot from eight yards was superbly beaten away by Marriott.
Boston replied with a long-range effort by Jason Kennedy but his shot was held by Rob Elliot.
On 34 minutes Boston went close to breaking through. Chris Holland's neat flick set Francis Green clear, but when he cut inside from the left his fierce shot was diverted away off Phil Edwards' head.
Boston were within a whisker of going ahead three minutes after the break. A quickly taken free-kick by Brad Maylett found Green in the clear. He chipped over Elliot but Edwards got back to hoof away from close to his own line.
There was another opening for the home side on 59 minutes when Mark Greaves fed Kennedy wide on the right, but his dangerous cross was scrambled away by the visitors.
Then in the 68th minute a swerving drive by Broughton almost caught Elliot unaware but the keeper responded well with a good save.
But Broughton wasn't to be denied for long. In the 72nd minute he side-footed Boston ahead from close range after Green had set up the chance with a low cross from the left.
Stanley stepped up the tempo in the closing stages and threatened in the 86th minute when a raking free-kick from the right by Whalley fizzed into the Boston area but Marriott held on well as the ball dropped into his grasp.