The Scotland winger stooped low to cancel out Darren Bent's earlier header and ensure the first match in this competition between the teams in 106 years ended in a 1-1 draw.
But Fulham also had goalkeeper David Stockdale to thank for a string of saves, including one close-range reflex stop from Norwich striker Ricky van Wolfswinkel.
The pre-match statistics had not been encouraging for Norwich. They had not recorded a victory over Fulham since March 1986, a 2-1 league success.
Since then the two sides had met 13 times in all competitions, with Fulham winning 10 of those encounters.
The most recent of those victories came on Boxing Day when Fulham snatched a crucial three points in their fight against relegation.
But with 17 changes in all from the sides these teams fielded in their Premier League matches on New Year's Day, eight from Norwich and nine from Fulham, there was an open, vibrant feel to the encounter.
It was Norwich who looked the more creative in the early stages, a Johan Elmander right-foot shot fizzing just wide and a Josh Murphy right-footer well smothered by Stockdale.
They have high hopes for Murphy at Carrow Road. The 18-year-old winger was making his full first-team debut and impressed with a series of mazy runs down the left, one shot from the edge of the penalty area, in particular, bringing a flying save from Stockdale.
There was no denying the work rate and determination on show, nor Norwich's domination. It was finishing quality which was in short supply, until, that is, the enigmatic Bent struck after 39 minutes.
The former England striker, on loan from Aston Villa and whose career has been littered with barren patches, latched onto a Chris David cross and rose above advancing goalkeeper Mark Bunn to head home his third goal in 13 matches for Fulham.
It was against the run of play and barely deserved but to their credit Norwich did not dwell on their misfortune.
In fact, they were level within five minutes and it was Murphy again who provided the inspiration. Another penetrating run saw him surge down the left before whipping in a precise cross which Snodgrass stooped to send past Stockdale.
They may not have been first-choice line-ups but, with Derek Boateng toiling in midfield for Fulham and Snodgrass a constant menace for Norwich, there was no doubt the FA Cup mattered at Carrow Road as the teams went in at half-time.
The high tempo continued in the second half and it was clear that neither manager wanted a replay to further clog up the fixture calendar.
Norwich would have taken the lead after 57 minutes if it had not been for the agility and reflexes of Stockdale.
A swinging corner from the left fizzed onto the head of Van Wolfswinkel and the Dutch striker's powerful header appeared goalbound until somehow Stockdale palmed it away at full stretch.
It was tough on van Wolfswinkel, the £8.5m purchase from Sporting Lisbon who was making his first start since October 6 following a troublesome toe injury and who so far has managed just one goal for Norwich.
Snodgrass might have snatched a second goal after 66 minutes but his swerving free-kick rippled the side-netting, although many in Carrow Road were already celebrating.
There was something of a unique occurrence after 79 minutes when Josh Murphy, making his full senior debut, was substituted by his identical twin brother Jacob, also making his full senior debut.
But while both teams had chances to seal their place in the fourth round they will have to fight it out again at Craven Cottage when the replay takes place the week after next.