Mohamed Salah and Nemanja Matic also found the net in the comfortable victory, though the Eredivisie side found time to pull a goal back in the dying minutes
The Spain international, who joined from Barcelona last month and made his debut in a 2-1 in against Olimpija on Sunday, curled in a delightful free-kick from 25 yards in Arnhem to give Chelsea fans a taste of the quality he possesses.
Chelsea were already one goal to the good by that stage, Mohamed Salah having converted Diego Costa's pass in the fourth minute as the visitors made a flying start.
Vitesse created opportunities, with Petr Cech called into action on several occasions in the first half, while Zakaria Labyad struck the post after the break.
The visitors always looked assured, however, and sealed victory when Costa provided his second assist of the game to tee up Nemanja Matic with 14 minutes to go.
Substitute Uros Durdevic pulled one back for Vitesse in the 89th minute, but that proved nothing more than a consolation.
Chelsea went ahead early on after Costa - who scored on his debut against Olimpija - sprung the offside trap and unselfishly squared to Salah, who had the simple task of tapping into an empty net.
Vitesse responded well, Valeri Kazaishvili thumping a powerful 25-yard effort that Cech did well to tip over the crossbar six minutes later, with Davy Propper glancing the resulting corner just off target.
Bertrand Traore, one of two Chelsea loanees in Vitesse's team, fizzed a low shot narrowly wide of Cech's left-hand post in the 15th minute as the hosts looked to recover from the early setback of going behind.
But Fabregas doubled Chelsea's advantage in fine style midway through the half, curling a free-kick over the wall beyond the despairing dive of Eloy Room into the top left-hand corner.
Kelvin Leerdam had Vitesse's best chance of the first half seven minutes before the interval, but he was brilliantly thwarted by Cech rushing from his line to block the shot.
Costa had an opportunity to put the result beyond doubt 14 minutes into the second period, but the ex-Atletico Madrid man was denied by the legs of replacement goalkeeper Piet Velthuizen.
Labyad almost brought Vitesse back into the match in spectacular fashion on the hour mark, cutting inside from the left and unleashing a swerving shot that beat Cech but crashed off the outside of the post and away to safety.
Matic duly added Chelsea's third in the 76th minute, profiting from more selfless work from Costa who raced clear and set up the Serbian midfielder to slot home.
It was the hosts who had the last say, Durdovic nodding home at the back post, but that did little to take the shine off an accomplished performance from Mourinho's side.