Playing in the competition for the first time since 2010, Everton could hardly have wished for a better start and were three goals to the good just after half-time.
An incisive passing move led to Ricardo Rodriguez scoring an own goal inside 15 minutes, while Seamus Coleman was also on target before the break and Leighton Baines netted a second-half penalty before a late Kevin Mirallas strike to mark his third in as many games.
Rodriguez went some way to compensating for his misfortune when he fired home a free-kick in stoppage time.
The margin of defeat was harsh on the visitors, who found goalkeeper Tim Howard in fine form, the American notably denying Luiz Gustavo, Daniel Caligiuri and Kevin De Bruyne in an inspired display.
But Everton were far more clinical and issued a statement of intent in Group H, maintaining their unbeaten European record against German opposition in the process.
Roberto Martinez said in the build-up that he would take the competition seriously and remained true to his word, naming the same side that defeated West Brom 2-0 in the Premier League on Saturday. Samuel Eto'o was fit enough for the bench.
The Bundesliga side controlled the early exchanges but Everton took the lead with an irresistible passing move.
Steven Naismith, Baines and James McCarthy combined beautifully to tee up the Scottish forward and, though his 12-yard shot was blocked on the line by Rodriguez, the ball ricocheted off the goalkeeper and in off the unlucky defender.
It was a move of such quality that the element of luck was perhaps earned, but Gustavo fired a warning shot two minutes later as he stung the palms of Howard.
Romelu Lukaku, who scored against his former club at the weekend, curled a 30-yard free-kick towards goal five minutes before the break but Diego Benaglio made a smart low save to deny the striker.
But Martinez's men did get their second in first-half injury time. Mirallas' rasping shot was beaten away by Benaglio but Baines reacted quickest to square the ball to his fellow full-back Coleman, who nodded in from six yards for his second goal of the season.
Wolfsburg needed to regroup but made a dreadful start to the second half.
Maximilian Arnold's loose pass was seized upon by Aiden McGeady who tumbled under the challenge of Robin Knoche and although initial contact was outside of the area, the referee pointed to the spot.
Baines made no mistake, lashing into the right corner after 46 minutes for his first of the campaign.
To their credit, Wolfsburg's response was positive and Howard had to be alert to save twice after 53 minutes.
De Bruyne's delicate pass found Caligiuri whose low effort was blocked by the American, who then saved Gustavo's rebound, though the midfielder ought to have done better.
And Howard made perhaps his best save 16 minutes from time when De Bruyne's curling shot was tipped wide at full stretch.
Everton extended their advantage in the 89th minute when Eto'o, introduced 20 minutes earlier, fed Mirallas to slot home with aplomb.
And though Howard was finally beaten in the last minute of injury time thanks to a superb Rodriguez free-kick from 25 yards, the night belonged to the hosts.
