The Welsh international grabbed his first goal of the day and for the club after just seven minutes after being set-up by Lubomir Moravcik.
French defender Bobo Balde doubled the lead right on the stroke of half-time.
Hartson then netted his second and his side's third on the hour with Moravcik again the provider.
The visitors pulled one back after 75 minutes with Jim McIntyre heading home Danny Griffin's cross, but Celtic made sure of the win when Hartson completed his hat-trick in the 83rd minute with a simple tap-in.
Young striker Shaun Maloney then hit the fifth two minutes later to round off a superb afternoon for Martin O'Neill's men.
The result moves Celtic 10 points clear at the top - for 24 hours at least - with Rangers visiting St Johnstone on Sunday.
Livingston moved to within one point of Rangers in second place with a superb 5-1 win over Kilmarnock at Rugby Park, despite going 1-0 down early in the game.
Loan signing Didier Santini put through his own net on his Livvy debut after just five minutes to give Killie the lead.
The lead was shortlived with David Bingham drawing the sides level just three minutes later.
Marvin Andrews then fired the visitors in front in the 20th minute to give them the lead at the break.
David Fernadez gave them a two goal advantage with a goal in the 68th minute and they wrapped up victory with a strike from Davide Xausa 10 minutes later.
Dario Aurellio netted the fifth with an injury-time penalty spot to give Jim Leishman's men a handsome victory.
Aberdeen moved into fourth in the table with a hard-fought 3-2 win over Dunfermline at Pittodrie.
Robbie Winters broke the deadlock after 21 minutes and Thomas Solberg doubled the lead just three minutes later.
Winters netted his second and the home side's third four minutes before the break from the penalty spot.
Scottish international striker Stevie Crawford pulled one back for The Pars in the 68th minute and Dutchman Jack de Gier grabbed a second for the visitors 10 minutes later, but they just could fine the equaliser in the final minutes of the game.
Dundee's game with Motherwell at Dens Park was postponed because of a waterlogged pitch.
Click on fixture for full match summary
Aberdeen 3-2 Dunfermline
Celtic 5-1 Dundee United
Kilmarnock 1-5 Livingston
Click here for updated league table
