Hearts played the most composed football but the loss of that early goal proved too much to overcome.
The visitors were unlucky when Andy Webster saw his goal ruled out for offside early in the second half and Hearts might have had a penalty when Bobo Balde kicked Roman Bednar.
There seemed little threat when Maciej Zurawski found Hartson 25 yards out but his first-time volley appeared to catch Craig Gordon off balance and it bounced past the keeper and just inside the post.
It was the start Celtic wanted but Hearts soon pressed them back. From Robbie Neilson's free-kick, Edgaras Jankauskas laid the ball back to Bednar and his shot looked to be creeping in before Artur Boruc got a vital touch.
Webster's shot just before the interval landed on top of the net and the Hearts defender thought he had scored after the restart when he touched home a Christophe Berra header but was ruled offside.
Boruc made another good save from Bednar with Paul Telfer completing the clearance.
The visitors' appeal for a penalty were waved away when Balde kiced Bednar rather than the ball with a wild swipe. Hearts knew it was not going to be their night when Dion Dublin headed off his own line and Michal Pospisil shot wide from close range.
Whilst this was no means a vintage Celtic performance there is no doubt that they are worthy champions as their massive 20-point advantage testifies.