Germany playmaker Havertz bagged his first Chelsea goals, relishing a more central role as Frank Lampard's men overpowered their Championship opponents.
Tammy Abraham, Ross Barkley and Olivier Giroud also found the net in a dominant Chelsea display at Stamford Bridge.
Toni Rudiger's omission for Chelsea's squad for the second match in a row left the Germany defender eyeing a loan exit before the October 5 transfer deadline.
Silva's arrival could push Rudiger down the pecking order in west London, with the Brazil stalwart adding immediate organisation in an authoritative hour's work.
Fikayo Tomori's return to the first team fold could compound Rudiger's situation, and the 27-year-old could now seek a move during this window.
Blues boss Lampard pledged to use this cup outing as a chance to boost fitness after an almost non-existent pre-season.
Chelsea strangled the Tykes with their high press, forcing a number of goals directly from turnovers.
Havertz took his goals well and looks most natural as a number 10, but the lessons proved precious few on a comfortable night.
Silva's calmness shone through right from the off, the Brazilian bossing his team-mates whether Chelsea were in or out of possession.
The 36-year-old's presence could not stop Barnsley creating a clutch of gilt-edged openings however, and the Blues needed veteran goalkeeper Willy Caballero to dig them out of several holes.
Abraham handed the Blues the lead though, pilfering the ball from Aapo Halme before applying a cool finish.
Mason Mount should have doubled Chelsea's lead having been played in smartly by Havertz, but could only drill wide from eight yards.
Havertz eventually did put the Blues 2-0 up, finishing in style after a slick move.
Abraham stepped over Mount's pass, and that dummy flummoxed the Barnsley defence. When the ball reached Havertz, he set himself and then delivered the killer blow.
Barnsley keeper Brad Collins was booked at the death of the half for upending Abraham just outside his area, and Chelsea took their two-goal lead into the break.
Barkley had Chelsea three to the good just minutes after the break, drilling low into the corner of the net after Havertz won the ball well on the edge of the Barnsley area.
Chelsea's press started to put the heavy squeeze on Barnsley, and quickly yielded another goal.
This time it was Mateo Kovacic who forced a turnover deep in Tykes territory, and Abraham laid off to Havertz to slot his second of the night.
Havertz completed his hat-trick by rounding the stranded Collins, latching onto Abraham's fine ball to put the seal on a hugely-promising individual performance.
Chilwell laid on the sixth goal for Giroud, capping his first outing with a smart assist to cap the Blues' stress-free night for.