Potters squandered a number of presentable first-half chances but still led 2-1 at the break courtesy of Charlie Gajic’s well-taken brace which sandwiched Fawzi Saadi’s looped effort for the hosts.
But Poore was enraged by referee Brian White’s decision to award Poppies a penalty when backpedalling full-back Dean Evans bumped into Connor Phillips while striving to win a header moments after the restart.
Marc Fairbrother levelled from the spot and to rub salt into the wound, Phillips later pounced on two hashed clearances from injured Verwood stopper James Collins to net a pair of tap-ins.
With fellow custodian Chris Lynch unavailable, Collins had tried to soldier on with a dislocated finger picked up during the game but was withdrawn and replaced by midfielder Gajic between the sticks after the fourth goal.
That change came too late, though, and the match petered out as Poppies claimed £1,500 in prize money and a preliminary round date at league rivals Bemerton.
Poore, however, was adamant that the earlier penalty decision had altered the course of events and went on to criticise White and assistants Michael Day and Jennifer Martin for their handling of the derby date.
“That decision completely changed the game and it frustrates me that someone in the middle has ruined the match in such a big competition," said Poore.
“Not once did Dean take his eyes off the ball, he has won the header but knocked over their lad while running backwards.
“If you watched 100 games you would rarely see that given and it has cost us because Poppies had a spring in their step after that.
“None of the three officials were strong enough for an FA Cup game and got a lot of decisions wrong both ways. It was irritating to say the least.”
Opposite number Ken Vaughan had sympathy over the call but still felt his charges deserved to progress.
The Poppies manager said: “We were fortunate with the penalty, it was very dubious and I would have been upset if it had been given against us but I didn’t feel we were lucky to win.
“Verwood worked very hard, especially in the first half but after half-time we were by far the better side.”
When approached by the Daily Echo, referee White politely declined to comment on Poore's remarks.
Poppies: Vaughan, Doe, Jacobs, Burgis, Pearce, Saadi, Hales (Ayrton, 51), Robinson (Cooper, 79), Bassiriki, Fairbrother, Phillips. Unused subs: Sullivan, Khider, Teixceira, Arthur.
Potters: Collins (Rutten, 74), Evans, Buck, Molyneaux, Dent, Gajic, Dyer, Cousins (Whelan, 51), Fisk, Groves, Foster (Goddard, 74). Unused subs: Anderton, Murray.