The Irishman completed his treble in the closing stages as United's convincing victory continued their strong revival from the difficult initial spell that followed Craig Levein's departure.
Only an injury-time goal from Pedro Moutinho denied them a fourth successive clean sheet as caretaker manager Peter Houston continues to lead them into a strong position in the race for Europe.
Falkirk manager Eddie May picked the same side that drew with Celtic the previous weekend, with Darren Barr, Colin Healy, Enoch Showunmi and Carl Finnigan all returning after being suspended or ineligible for Monday's Scottish Cup defeat at Kilmarnock.
A draw would have taken the Bairns off the bottom but they were sluggish from the opening stages, when several poor clearances put them under pressure.
However, the hosts had a great chance in the 12th minute when Showunmi got to the byline and crossed for Carl Finnigan four yards out, but the striker's header hit goalkeeper Dusan Pernis at point-blank range.
Darren Dods replaced the injured Andy Webster after 14 minutes but United continued to control possession on a bumpy surface before Goodwillie notched his goal.
The Scotland Under-21 striker charged down Brian McLean's attempted clearance on the edge of the box and held off the defender's sliding tackle before rolling the ball through Robert Olejnik's legs.
United midfielder Danny Swanson fired an angled drive just wide from 25 yards before Showunmi had a shooting chance from a corner, only to volley three yards over with his left foot.
Falkirk lost McLean to injury in the 40th minute, with Lee Bullen coming on at right-back and Barr moving into the centre of a defence which was exposed in the 45th minute.
Pernis launched a huge kick after saving Ryan Flynn's weak 20-yard effort and Daly easily shrugged off Pedro Pele before lobbing Olejnik from 20 yards.
May made his final substitutions at the break with Jack Compton and Moutinho on for Scott Arfield and Finnigan.
Daly had the ball in the net again four minutes into the second half after challenging Olejnik at a second ball following a free-kick, but referee Euan Norris had spotted an infringement.
Compton sparked Falkirk into life with a powerful volley from the left edge of the box, which Pernis pushed into the path of Pele, whose delayed follow-up was blocked.
Showunmi then headed over Compton's corner but Falkirk's comeback attempts were undone in a 57th-minute counter-attack. Kenneth's challenge on Showunmi allowed Swanson to break and he set up Daly to slide home from 10 yards.
Moutinho blazed over from inside the box but Falkirk were struggling to put United under pressure.
Damian Casalinuovo replaced Goodwillie and found room inside the box to get away a shot that Olejnik blocked with his feet.
Healy had a powerful shot blocked by Pernis but United continued to look the better side going forward.
Olejnik saved at close range from Dods and Daly before the Irishman headed in Swanson's left-wing cross in the 88th minute.
Moutinho then nodded home Compton's cross but there were few Falkirk fans left inside the stadium to see their consolation.
Source: Team Talk