Nick Viergever collected Ruben Ligeon's 39th-minute pass to break the deadlock and Denmark midfielder Lasse Schone doubted the hosts' advantage with a free-kick three minutes into the second half.
Mike van der Hoorn headed a third in the 62nd minute after Guram Kashia hit the bar for Vitesse.
The visitors managed to get on the scoresheet after 84 minutes through Marko Vejinovic but Schone ensured a three-goal victory for Ajax by completing his brace - taking Frank de Boer's side top after just one game.
Netherlands international Memphis Depay brought his impressive World Cup form into the opening weekend as PSV Eindhoven won 3-1 at 10-man Willem II.
Goalkeeper Kostas Lamprou endured an afternoon to forget as the home team marked their return to the top flight.
Lamprou was sent off with 15 minutes remaining, bringing down Georginio Wijnaldum for Depay to slot his second from the penalty spot, having missed a seventh-minute clearance that allowed Jurgen Locadia to fire into an empty net.
The Willem goalkeeper found himself wrong-footed when Depay drove at his defence and drilled into the bottom corner three minutes later.
Bruno Andrade finished smartly on the turn in the 40th minute and Willem threatened a comeback, but it was not to be for the Eerste Divisie champions.
Feyenoord also benefitted from playing against reduced opposition, although they had to wait until the 90th minute for substitute Mitchell te Vrede to score the only goal against ADO Den Haag.
ADO gained control as the first half progressed but they were dealt a body blow as Mitchell Schet picked up a second yellow card in the 40th minute.
Feyenoord struggled to find a way through but belatedly landed a sucker punch when Te Vrede headed Jordy Clasie's free-kick into the corner.
Elsewhere, FC Groningen secured a stunning 3-2 comeback win over Go Ahead Eagles.
Alex Schalk's inventive backheel from Deniz Turuc's shot gave the Eagles a 19th-minute lead and Lesley de Sa cut in from the right flank to send a rasping shot into the top corner early in the second half.
Tjaronn Chery reduced the arrears from the edge of the box and it was all square with 15 minutes to play as Michael de Leeuw headed in from Jarchinio Antonia's floated centre.
And ex-Go Ahead winger Antonia broke the Deventer faithful's hearts with a diving header of this own in the 87th minute.
