The 29-year-old, who completed his move from Arsenal on Monday, played a key part in Marcus Rashford's opener before setting up Ander Herrera to score the second at Huish Park.
Substitutes Jesse Lingard and Romelu Lukaku added late goals to cap a solid performance from United, who have now won five games in a row in all competitions without conceding.
Manager Jose Mourinho had promised to play a strong side and a number of senior players were involved, with Sanchez joining Juan Mata and Rashford in attack and club captain Michael Carrick handed his first start since September.
Yeovil, who are 21st in England's fourth tier, acquitted themselves admirably for the first 40 minutes but were undone when Rashford capitalised on some uncertain defending after being set up by Sanchez.
United's control rarely relented from then on and Herrera finished a clinical move for the second goal on his 150th appearance for the club, with Sanchez claiming the assist.
The Red Devils' new number seven was taken off after 72 minutes, having taken a few knocks in a combative first outing for the club, before Lingard's fine run and finish and Lukaku's tap-in ensured United progressed comfortably to round five.
15 - Alexis Sanchez has been directly involved in 15 goals in 15 appearances in the FA Cup (eight goals and seven assists). Impression. pic.twitter.com/pg6IKokbT4— OptaJoe (@OptaJoe) January 26, 2018
A bright start from Yeovil saw them create the first two chances of the match, Jordan Green shooting straight at Sergio Romero from a tight angle and Omar Sowunmi sending a tame header into the goalkeeper's gloves from a good position.
Sanchez, who was booed by a small section of the home fans from the start, looked eager to get involved from the off and set up Scott McTominay for United's first chance after 25 minutes, with Artur Krysiak making a good low save to his right.
Sanchez had two further efforts before the break, sending a free-kick into Krysiak's hands before dragging a shot wide following Mata's pass, but Yeovil were competing well to keep United to half-chances alone.
That was until the 41st minute, when some slack defending proved the home side's downfall. Rashford lost the ball just inside the penalty area but, as Tom James tried to shepherd the ball back to Krysiak, Rashford stole in behind him and side-footed into the net.
4 - Marcus Rashford is the fourth different @ManUtd player to score 10 or more goals in all competitions in 2017/18 (with Lukaku, Lingard and Martial); @ManCity are the only Premier League team with as many (also four). Riches. pic.twitter.com/vgG9uY5i3R— OptaJoe (@OptaJoe) January 26, 2018
With the lead secured, United pushed for a key second goal early in the second half, Rashford seeing a shot deflected wide after a good run before Mata turned the resulting corner onto the top of the crossbar.
Yeovil responded by forcing a corner of their own but they were undone by a slick counter-attack from the visitors in the 61st minute. Mata released Sanchez down the left after United had cleared, the Chilean found Herrera just inside the penalty area, and the midfielder, who scored on this ground in a cup clash in 2015, drilled a left-foot shot across Krysiak and into the bottom-right corner.
Mata thought he had scored his side's third from Matteo Darmian's effort across goal but the offside flag denied him, before Rashford was thwarted by Krysiak, with Sanchez withdrawn shortly after pulling up with a knock.
United controlled the closing stages and Lingard made certain of their place in the next round, cutting infield from the right before firing low into the left-hand corner past a static Krysiak.
Fellow substitute Lukaku put the seal on a happy 55th birthday for Mourinho in his 100th game in charge, steering Marcos Rojo's cross from the left high into the net with practically the last kick of the game.
Key Opta stats:
- Manchester United have progressed from 13 of their last 14 ties in the FA Cup fourth round, with their only defeat in this run coming back in 2012 (1-2 v Liverpool).
- This was Jose Mourinho's 100th game in charge of Manchester United – he has won 62 per cent of his games since taking over at the start of last season (P100 W62 D23 L15).
- The Red Devils have now won each of their last five games; their longest run of victories in all competitions since winning six in a row in September 2017.
- Zlatan Ibrahimovic (29) is the only Manchester United player who has scored more goals for the club under Jose Mourinho than Marcus Rashford (21).
- Alexis Sanchez has been directly involved in 15 goals in 15 appearances in the FA Cup (eight goals and seven assists).