The midfielder got the hosts off to a flying start with the opening goal inside the first two minutes before adding a second six minutes later from a Willian corner.
Loftus-Cheek then sealed his hat-trick early in the second half, as he fired a low shot past goalkeeper Denis Scherbitski.
Alexei Rios grabbed pulled one back for the visitors late on.
The result means Chelsea remain in first place in Group L. The south-west London club lead the way with maximum points from their opening three games and are six points clear at the top.
Loftus-Cheek Must Push for Loan Move After Performance
There's no doubt that Loftus-Cheek was Chelsea's star man with his three goals. The 22-year-old started for just the second time this season and made the most of a rare opportunity to impress manager Maurizio Sarri.
His first goal was a tidy first-time finish from a David Zappacosta cross. Goal's Nizaar Kinsella noted how it was his first Chelsea goal in over two years:
His second goal came from close range and put Chelsea in complete control with less than 10 minutes played.
However, it seemed that he was still not doing enough to completely satisfy Sarri:
Loftus-Cheek scored again with a fine curling strike from outside the box to seal his first professional hat-trick:
Yet despite his goals and match-winning performance, Loftus-Cheek still faces a tough challenge for minutes in the Chelsea team, let alone a regular place.
Sarri has plenty of midfield options with N'Golo Kante, Jorginho, Mateo Kovacic and Ross Barkley all ahead of Loftus-Cheek in the pecking order.
His performance against BATE will surely have enhanced his first-team chances. However, if he wants to play regularly this season, he'll need to push for a loan move away from the club in January.
Loftus-Cheek spent last season on loan at Crystal Palace, where his performances saw him secure a place in the England squad for the 2018 FIFA World Cup.
The midfielder became a key player for the Eagles, and a similar move away could help continue his development, which will only stall on the sidelines at Stamford Bridge.
Zappacosta Deserves to Start Ahead of Azpilicueta
Zappacosta is another player who took the chance to impress on a rare outing from the start.
The 26-year-old set up Loftus-Cheek for his first goal, provided plenty of quality delivery into the box from the right flank and also tested goalkeeper Scherbitski with a fierce shot.
The Italian brought energy and end product to the right-back position, which has been lacking this season with first choice Cesar Azpilicueta out of form.
Matt Somerford of the Press Association has said Azpilicueta is in decline:
In contrast Zappacosta looked bright out wide on the right flank, and his ability to run and cross could also benefit Chelsea's misfiring strikers, Olivier Giroud and Alvaro Morata.
Giroud has now gone six games without a goal for the Blues, while Morata has managed just three all season.
As an attacking full-back, Zappacosta's defending has been called into question at times. But he will not be allowed to get away with that under Sarri, as noted by Kinsella:
Zappacosta may not be the perfect option at right-back for Chelsea, but he showed he deserves a chance to stake a claim in the first team.
Chelsea are in Premier League action on Sunday when they head to Turf Moor to take on Burnley. BATE also play on Sunday; they are away at Gomel in the Belarusian Premier League. The two teams will face off again in Europa League action at Borisov Arena on Nov. 8.