But Everton's third-round place came at a price. Already blazing a trail at the top of the table for red and yellow cards, they had Materazzi sent off late in the game. The Italian defender, cautioned for dissent in the first half, left Mr Winter no option after clattering into Marcus Stewart.
On the day when Everton lavished a record fee on a goalkeeper, Steve Simonsen, it was just Thomas Myhre's luck that his first touch would be to fish the ball out of the net.
Not that the Norwegian was at fault - Huddersfield had been allowed to build down the right and the defence were static as Stewart converted Paul Dalton's cross for his eighth goal of the season.
The move was initiated by Barry Horne, the former Everton captain, after a characteristically biting tackle and pass. The player who has inherited his mantle in midfield, John Collins, was one of three regulars left on the bench, suggesting that Walter Smith places a rather higher priority on Premiership safety.
Stewart might have doubled Huddersfield's lead within three minutes. As he steadied himself, after again being played in by Horne and Dalton, David Unsworth made a vital saving tackle.
Unsworth also cleared Steve Jenkins' header off the line in the 20th minute, and the reprieve served to stir Everton. The equaliser arrived in the 29th minute. Following Sam Collins' crude challenge on Don Hutchison, Dacourt powered the free-kick into the left-hand corner of Nico Vaesen's net from 25 yards.
Three minutes before half-time, with Everton's pressure mounting, Materazzi back-heeled their second goal after Vaesen had made a poor fist of punching a corner.
Materazzi was entitled to feel that justice had been done. When Huddersfield were leading, he headed a "goal" which was disallowed for offside, presumably against Duncan Ferguson, though the Scot was hardly interfering with play. It was noticeable, however, how the decision, and the angry response it sparked from players and spectators, also helped to galvanise Everton.
The drama took a further twist in the 70th minute. No sooner had John Oster skied the ball over the gaping Huddersfield goal from five yards, than the referee went back to the Everton half and dismissed Materazzi for an earlier foul, leaving Myhre to face an unexpectedly testing finale.
Everton (3-5-2): Myhre; Unsworth, Materazzi, Tiler; Cleland, Dacourt, Hutchison, Oster (Collins, 76), Ball; Cadamarteri (Barmby, 84), Ferguson. Substitutes not used: Milligan, Watson, Gerrard (gk).
Huddersfield Town (3-5-2); Vaesen; S Collins, Morrison (Barnes, 73), Gray; Jenkins, Dalton (Beresford, 73), Horne, Johnson, Edwards; Stewart, Thornley. Substitutes not used: Dyson, Browning, Francis (gk).
Referee: J Winter (Stockton-on-Tees).