Newcastle took the lead in the 11th minute with a goal out of nothing.
They extended their lead in the 21st minute and again Bolton's defending was poor.
James Milner was able to weigh up his cross and found Obafemi Martins at the back post. He beat Jaaskelainen with a superb overhead kick.
The agony continued for Bolton in the 27th minute when Martins made it 3-0 with his second goal of the game.
He had a bit of luck, however, his shot from 20 yards taking a deflection off defender Abdoulaye Faye to leave Jaaskelainen stranded.
Nicolas Anelka pulled a goal back for Bolton in the 50th minute but Newcastle were never troubled again.
Considering Allardyce was without a number of key players, he had good reason to feel satisfied.
He said: "It has been a great day for me, bringing my side back to my old club.
"Your hope your players produce a committed performance, not just as individuals but as a team.
"That happened in the first 45 minutes and we killed the game off stone dead.
"Bolton put us under the cosh a little bit but we saw the game out extremely well."
Bolton manager Sammy Lee, who succeeded Allardyce, was at a loss to explain their start.
He said: "I do not know what went wrong. We did not turn up for the first 45 minutes.
"We wanted to get the ball down and play but we did not do that and it was disappointing.
"But I got a response in the second half. However the over-riding thing was how poor we were early on."