Woolford, 25, struck in the 14th minute with a clinical left-foot finish to register his seventh goal of the season.
He then turned provider for Dagnall to claim a killer second with six minutes remaining and seal a superb win for Ian Baraclough's men, who have done no favours for Laws.
The former Scunthorpe manager had guided the Clarets to victory at Barnsley on Boxing Day but faced criticism from the home fans after this dismal defeat left them outside the play-off places.
Burnley's win at Oakwell two days previously marked their first away win of the season, prompting Laws to name an unchanged starting line-up to face Scunthorpe.
The visitors arrived at Turf Moor in the relegation zone, yet they began in vibrant fashion and quickly built up a head of steam.
Although Burnley striker Jay Rodriguez headed just wide in the third minute, Scunthorpe soon gained control and striker Jonathan Forte drilled a 20-yard shot narrowly over the bar.
Two minutes later, Woolford also went close from distance and then Michael O'Connor had a 20-yard shot well parried by home goalkeeper Lee Grant.
But the Iron were not to be denied and they found the net in the 14th minute when Woolford swept home a fine left-footed shot from 12 yards.
Midway through the first half, Burnley sought to find an equaliser but Rodriguez headed wastefully wide from 12 yards out.
Clarets winger Ross Wallace then headed goalwards from 12 yards but Iron goalkeeper Joe Murphy made a comfortable save.
The pendulum swung back in the visitors' favour moments later as Andrew Wright saw his 25-yard shot blocked.
Jack Cork, on loan at Burnley from Chelsea, then headed straight at Murphy from 12 yards and Scunthorpe could have doubled their lead seven minute before the break.
Forte found the space to strike a shot goalwards from 12 yards but Grant made a fine parried save to avert the danger.
As the interval approached, Burnley threatened from a corner but Clarke Carlisle could only head over the bar from Wallace's right-wing delivery.
Laws made a tactical switch at half-time, replacing the ineffective Wade Elliott with the more adventurous Chris Eagles.
The change looked promising as Eagles quickly got onto the ball and began teasing and tormenting the Scunthorpe defence before delivering a series of dangerous crosses.
In the 48th minute, Burnley's pressure almost told when Rodriguez headed over from Wallace's cross from just six yards out.
Forte then flashed a shot just wide at the other end of the pitch as Wallace and then Woolford also went close.
Laws replaced on-loan Manchester City striker John Guidetti, 18, with Chris Iwelumo in the 55th minute as the hosts looked to adopt a more physical approach.
Eagles saw a fierce free-kick saved by Murphy in the 62nd minute before going close again with another long-range strike in the 76th minute.
Carlisle then headed wide before Scunthorpe sealed the points when Woolford showed intelligence to find Dagnall and he rifled home from six yards out.
Source: DSG