Enzo Maresca's side had taken two points from an available nine -- against Arsenal, Leeds and Bournemouth in their previous three domestic outings -- and came into the game on the back of a potentially damaging Champions League defeat at Atalanta in midweek.
Conversely, David Moyes' side had four wins from five and, despite a 31-year winless league run at the Bridge, were the better team for large portions of the first period.
While Chelsea enjoyed fleeting spells in the first-half ascendancy, both goals came against the run of play. With England manager Thomas Tuchel watching on, Palmer demonstrated his value to club and country with a fine opener on 20 minutes, calmly drilling home after an astute Gusto pass. Then, despite the visitors arguably being worthy of parity, the France defender doubled the hosts' lead on the stroke of half-time, prodding in to cap a fine move spearheaded by Pedro Neto.
A strong start to the second period from the hosts might have yielded further daylight on the scoreline but Everton's resistance gradually waned and, although the lively Jack Grealish passed up an inviting opportunity with 20 minutes to go and Iliman Ndiaye struck a post late on, Chelsea were able to resume their top-four challenge in relative comfort.
Positives
Fundamentally, arresting the winless run was key. Beyond that, a fit and firing Palmer back amongst the goals can only be good news for the Blues, while Pedro Neto's fine season continues apace and right back Reece James's versatility compensated for the absence of suspended midfielder Moisés Caicedo.
Negatives
The Blues were lucky there was wastefulness from the visitors at times, while that same issue afflicted winger Alejandro Garnacho at the other end. Alongside him, Joao Pedro's quiet afternoon raises concerns over Chelsea's focal point in attack, especially with Liam Delap on the sidelines.
Manager rating out of 10
Enzo Maresca, 7 -- The boss guided his side back to winning ways at a key juncture in their season and his personnel changes were central to the result: the returning Gusto notched both a rare goal and an assist, while Palmer's deployment allowed the hosts to gain an early foothold in proceedings.
Once 2-0 up, the Italian made sure his side managed the outcome comfortably.
Player ratings (1-10; 10 = best, players introduced after 70 minutes get no rating)
GK Robert Sánchez, 7 - A welcome return to form after a shaky midweek display. He made two key interventions during Everton's first-half dominance, denying Thierno Barry a tap-in and thwarting Grealish in a one-vs.-one, and he would have embellished his display with a fine assist had Garnacho not finished wastefully.
DF Malo Gusto, 8 - Returned to starting XI with an assist inside 20 minutes, producing an astute ball for Palmer. While Grealish posed a stern test defensively, Gusto continued to threaten in attack, notching only his second goal for the club to ensure the win.
DF Wesley Fofana, 7 - Recovered from injury to play, and communicated well with the rest of his backline to keep Everton's forwards quiet. Produced a couple of important interceptions, but also received a late booking for a cynical foul on James Garner.
DF Trevoh Chalobah, 7 - Like Fofana, he looked sharp and communicative for the most part. He managed the physical threat of Barry -- and his replacement, Beto -- with little fuss and produced a key late block from Carlos Alcaraz to preserve the clean sheet.
DF Marc Cucurella, 6 - Started sharply, and had to tidy up more than once down the Everton right, but had a generally quiet game with most of the key moments emerging on the opposite flank.
MF Reece James, 7 - Wherever he plays, the captain is dependable and he filled in ably in midfield. In addition to a tidy, understated display, James provided his customary threat from set-piece delivery and almost notched a third for Chelsea, seeing his free kick well saved by Jordan Pickford.
MF Enzo Fernández, 7 - Occasionally sloppy in possession during the first period, Fernández grew in stature as the game progressed. He should have been rewarded with an assist for Garnacho after putting in a sublime cross which his compatriot spurned, while he also had a late shot of his own blocked.
FW Pedro Neto, 8 - Played a key role in the hosts' killer second goal by turning on the afterburners, and he generally tormented Vitalii Mykolenko whenever he gained possession. The Portugal winger's fine form continues unabated.
FW Cole Palmer, 8 - A welcome return to the starting XI. Palmer underlined his worth with a brilliant opener, just as the home crowd were showing unrest, and he crammed a neat but eye-catching performance into just under an hour on the field.
FW Alejandro Garnacho, 6 - Back in the starting XI, the Argentina international started on the front foot and seemed anxious to add to his haul of two goals this season. That desperation contributed to the spurning of an open goal and his defensive work was mixed, but he generally posed Everton problems.
FW João Pedro, 6 - Another largely quiet afternoon for the Brazil international, who was well managed by the visiting defense. He dropped deep -- sometimes too deep -- to link the play, but looked increasingly frustrated as the game wore on. He badly needs to add to his haul of four Premier League goals.
Substitutes
Andrey Santos (Palmer, 58'), 6 - Started and led one particularly promising counter-attack after his introduction, but blazed his shot well over the bar.
Jamie Gittens (Garnacho, 65'), 6 - He might have rounded off the afternoon with a last-gasp third, but undid the move with a dragged finish.
Estêvão (Joao Pedro, 81'), N/R - A popular introduction for the home fans, but he had little time to impact proceedings.
