The Spanish striker rescued his side by netting just his second goal of the season, and his first in seven games, to break down the visitors' resolute defence.
Victory was all that mattered in the end, however, and the three points take Chelsea clear at the top of Group L ahead of their double-header against closest challengers BATE.
The hosts' first clear-cut opening fell the way of matchwinner Morata, who slotted wide from a one-on-one position once slipped through by compatriot Pedro.
Davide Zappacosta was wasteful with a couple of attempts on his return to the side - one of eight changes from the Blues' 1-1 draw with Liverpool at the weekend - although he did force Tomas Tujvel into a routine stop with one of them.
Tujvel was also needed to keep out Pedro, as a frustrated Chelsea - now unbeaten in 10 games under Sarri since the Community Shield - all too often resorted to shots from range.
Chelsea felt that they should have had a penalty when Ruben Loftus-Cheek's run was halted by Paulo Vinicius, but Vidi had an even bigger shout rejected five minutes later as Gary Cahill cut across Boban Nikolov.
Loic Nego then slotted a shot wide when capitalising in a breakdown in communication at the back for Chelsea, before forcing Kepa Arrizabalaga into his first stop in a strong end to the first half for the visitors.
That momentum was taken into the second period as a Videoton side competing in Europe for the 10th time this season carved out a glorious chance for Nego, which Kepa was equal to down low.
It was a compliment to Vidi that Sarri was forced to turn to Hazard from the bench with just 11 minutes of the second half played.
Even the introduction of the in-form Belgian could not inspire Chelsea at first, though, and Vidi saw a couple of good openings come and go for Georgi Milanov, including a wayward shot from eight yards.
The arrival of Ross Barkley 10 minutes or so later had more of an effect, as the English midfielder tested Tujvel almost instantly with a long-range drive.
Having held out for so long, Videoton were undone with less than a quarter of the match left as Willian flicked Cesc Fabregas's clipped pass into the middle, where Morata was waiting to pounce with a good finish from close range.
Chelsea looked a lot more comfortable for a large majority of the remainder of the match, coming close to a quickfire second through Barkley's header from a Hazard delivery - the crossbar coming to Vidi's rescue.
The three-time Hungarian champions could easily have snatched a late equaliser, however, as a big save from Kepa was required to push aside Istvan Kovacs's outside-of-the-boot shot from the last meaningful act of the match.
CHELSEA (4-3-3): Kepa; Zappacosta, Cahill, Christensen, Emerson; Fabregas, Loftus-Cheek (Barkley 66'), Kovacic; Pedro (Hazard 54'), Morata, Willian (Moses 74')
VIDEOTON (4-2-3-1): Tujvel; Vinicius, Fiola, Juhasz, Stopira; Hadzic (Patkai 83'), Huszti (Kovacs 77'), Milanov (Hodzic 83'); Nego, Nikolov, Scepovic