Poyet was on the spot again to head home Simon Davies' cross after 23 minutes.
He's just a goal away from double-figures already and Glenn Hoddle could, arguably, talk of the bargain of the season having secured his signature from Chelsea for just £2million.
Les Ferdinand added a second goal after 51 minutes, his 13th of the campaign so far, and a home tie against Bolton in the fourth round was secure.
Glenn was forced into one change from the side that started against Ipswich Town on Saturday with Simon Davies stepping in wing-back in place of injured Christian Ziege. Les Ferdinand, who began that game but went off after 20 minutes with an ankle problem, was fit to start.
It was one-way traffic in the first half as City, heavily reliant on ex-Liverpool midfielder David Thompson, were restricted to shots from distance, all gathered with minimum fuss by Neil Sullivan.
Davies, continually pecking away at Antonelius down the left flank, had the first effort at goal, a glancing header from Sergei Rebrov's cross after 12 minutes and after Thompson's shot whistled wide, we opened the scoring.
Darren Anderton picked up the ball on the left flank and crossed deep, Davies picked the ball up on the far byline, crossed and Poyet was there to head past Marcus Hedman.
Ferdinand curled inches wide two minutes later but City responded with a break from our corner on the half-hour as Thompson picked out Julian Joachim's angled run but the striker failed to seriously test Sullivan.
The next chance fell to Ledley King who burst forward onto Tim Sherwood's perfectly-weighted pass only to look square for Ferdinand when an attempt on goal was on. Ferdinand closed the half with a textbook header just over.
It didn't take long for Ferdinand to make his mark though. The second half was only six minutes old when Poyet broke forward and held off Breen long enough to feed Anderton on the right. His low, teasing cross to the near post was turned onto his own post by Breen and Ferdinand poked the ball home from a yard.
Coventry rallied briefly as Carsley and Joachim fired wide but our dominance was almost total by the hour mark as Ferdinand's header was clawed out by Hedman and Sherwood volleyed wide.
Sherwood then somehow failed to finish off one of the moves of the season. We kept the ball for a good couple of minutes before a flurry of one-touch passes released Anderton. His pinpoint cross from the byline found Sherwood no more than two yards out but he ballooned his shot over the crossbar.
Taricco curled just over, Rebrov twice shot straight at Hedman and substitute Steffen Iversen blasted over from 18 yards.
In fairness the result was never in doubt - we won 'at a canter' one reporter read over to his office - just what the doctor ordered after our last couple of weeks.
Gus Poyet suggested at the start of the week that this was the start of a seven-day, three-match period that could set us on our way for the rest of the season. How apt then that he should ensure that it was one down, two to go at Highfield Road.