However, the result was very harsh on the visiting side who contributed much to an entertaining game.
Indeed it was only some fine saves by Artur Boruc and the woodwork which denied Thistle a consolation goal, although Celtic were undoubtedly worthy winners.
First-half goals by Jan Vennegoor of Hesselink and Massimo Donati were added to in the second half by Shunsuke Nakamura and Hesselink again. The luckless Phil McGuire conceded an own goal to round off a miserable result for the hapless visitors.
Thistle looked sharp in the early stages and wasted a good chance when Don Cowie's ball from the edge of the area was hooked over by Dennis Wyness from six yards.
Celtic took the lead on the quarter-hour mark when a flowing move involving Gary Caldwell and Nakamura ended with Hesselink deflecting the ball home with his knee from Scott Brown's cross.
Thistle took this setback in their stride and continued to advance on the Celtic goal with some clever play.
Grant Munro headed wide from a corner and Marius Niculae shot wide. Barry Wilson tested Boruc with a powerful drive before the Celtic keeper pulled off a fabulous save to touch away Cowie's net-bound shot.
It was very much against the run of play when Celtic scored a second just before the interval. Donati controlled Aiden McGeady's pass before cutting past Ross Tokely and shooting home from an acute angle.
McGeady continued to impress in the second half and some wonderful wing play ended with Scott McDonald heading into the side-netting.
Celtic added a third with a marvellous strike. Nakamura latched onto a loose ball 25 yards out, looked up and then dispatched a superb left-foot shot into the top corner.
Worse was to follow for the visitors when Hesselink was left with a simple header to the back post to notch Celtic's fourth.
Thistle were unlucky when Niculae controlled Tokely's cross on his chest before hitting his shot against the post and from the rebound Boruc did well to block Roy McBain's shot.
Celtic made it five in the 70th minute when McDonald's cross was turned into his own net by McGuire.