Michael Tonge scored the only goal in first-half stoppage time, making the most of one of the mistakes which are all too common in the Burnley rearguard.
United were limited to a clutch of long-distance strikes until goalkeeper Jensen's fluffed clearance fell to Tonge 18 yards out, who could hardly fail to miss an open net.
The visitors, whose defensive record on their travels was superior only to bottom club Wimbledon in the First Division prior to this game, blunted United's threat without creating much themselves before Tonge broke the deadlock.
With both teams missing arguably their most influential players - a foot injury kept Michael Brown out of the United line-up, while Burnley's Glen Little was sidelined by flu - there was a shortage of clear-cut opportunities in the opening 45 minutes.
Richard Chaplow, Burnley's scorer in last weekend's draw against Cardiff, tried his luck with a long-range strike which was comfortably saved by Paul Gerrard.
Mo Camara and Arthur Gnohere were grateful that their misjudgments in the Burnley penalty area were not punished by Ashley Ward, and Jensen watched Peter Ndlovu's shot fizz wide of his left upright shortly after.
The Clarets goalkeeper was not unduly tested by Tonge's drive from 18 yards, although he had to be alert to keep out another Ndlovu effort.
Chris Armstrong flashed a shot wide - Chaplow's header over the crossbar was the closest Burnley came to breaking the deadlock - before Jensen's blunder fell at the feet of Tonge, who made no mistake from the edge of the penalty area.
Luke Chadwick and Gareth Farrelly responded for Burnley early in the second half, and Gerrard was given another scare by Ian Moore just before the hour mark.
Jack Lester wasted the best chance of the half, however, when he lifted his close-range shot over the crossbar from Ndlovu's cross.
Lester was soon replaced by Paul Peschisolido, who saw his downward header from Rob Kozluk's cross superbly scooped away by Jensen.
Though showing commendable application, Burnley's lack of invention meant they rarely looked likely to grab an equaliser.