With both sides leaving their shooting boots back in the dressing rooms, the gritty goalkeepers repelled everything that was thrown at them to the frustration of the sell-out crowd.
Fourth-placed City had already won at Upton Park on the opening day of the season and, following Wednesday night's victory at Newcastle United, Sven-Goran Eriksson made just one enforced switch as the unwell Elano made way for Mexican debutant Nery Castillo.
Injuries had taken their toll on Alan Curbishley's squad over Christmas and, after losing at table-topping Arsenal on New Year's Day, the Hammers boss made two changes.
The injured Freddie Ljungberg was replaced by Matthew Etherington, while Dean Ashton returned in place of Jonathan Spector.
Having played four games inside the past fortnight, both sides were showing similar signs of a festive hangover as Matthew Upson, Etherington and Ashton sent half-hearted efforts well wide.
Meanwhile at the other end, Stephen Ireland's hopeful lob cleared the target and man-of-the-match Martin Petrov pursued referee Rob Styles after his penalty appeal was waved away.
The tie finally burst into life midway through the half when Green diverted Ireland's rising eight-yard shot high into the chilly East End skies after Petrov strode deep into home territory.
At the other end, Hart showed equal prowess when he clawed away Etherington's sizzling 12-yarder from the foot of his right-hand upright and, shortly afterwards, the in-form City keeper then fisted away Mark Noble's 18-yard free-kick from his near post.
As the half drew to a close, Castillo almost marked his debut with a goal but, with Anton Ferdinand snapping at his heels, he spooned Petrov's dangerous low cross into the six-yard box high over the bar to keep it all-square at the interval.
Curbishley replaced skipper Lucas Neill with Spector for the start of a second half that soon saw Green race out to bravely pluck the ball off Vedran Corluka's toes, before Ireland again forced him to divert aside an angled rocket that was then hacked clear.
With the game heading for an inevitable replay, both managers shuffled their packs as substitutes Kelvin Etuhu, Gelson Fernandes, Rolando Bianchi, Kyel Reid and Lee Bowyer all entered the proceedings for the final stages.
But after Micah Richards saw his diving header blocked and Petrov tickled the bar with a deft chip, Hart earned his side a deserved replay when he tipped Ashton's close-range header over the top.