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But after failing to cash in on two outstanding chances in the opening minutes they never again looked like troubling Richard Wright in the Everton goal.
For Everton, who in past years have had their own relegation battles, it was a heady return to the upper reaches of the Premiership as Steve Watson's two first-half goals virtually sealed the game.
Bolton's only consolation was a stoppage-time goal from Delroy Facey but Allardyce's side looked in need of new faces before the transfers window slams shut.
Bolton could have been two goals up in the opening four minutes.
Transfer-listed Michael Ricketts could have done his hopes of a move a power of good had he managed to get a header on target after only 30 seconds while a clever free kick by Youri Djorkaeff ended with Ivan Campo glancing it a foot wide.
That, however, failed to rattle Everton as they responded by having the better of the half with Gary Naysmith and Brian McBride both firing in warning shots.
The strong swirling wind proved difficult for both sides but Everton's better passing game often worried a jittery looking home defence.
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Bolton tried to change things at half time with the introduction of Kevin Nolan but it was still Everton who called the shots and Jaaskelainen had to make a tremendous save to tip a goalbound effort from Naysmith over the bar.
Djorkaeff and Per Frandsen both fired in long-range efforts as Bolton struggled to get back into the game but the danger always came from Everton.
They could have increased their lead had they taken the chances they created as the home side left gaps at the back in a desperate search for a goal.
Even so Watson should have completed his hat-trick in the 77th minute but saw his close-range header well saved.
In stoppage time Bolton did manage to break down the Everton defence as Facey fired the ball home after Frandsen's shot had been blocked.