Swindon threatened Mark Schwarzer's goal on a number of occasions and Andy Johnson failed to convert a second-half penalty on his first start since October.
Fulham boss Roy Hodgson was looking to earn his first win since signing a new 12-month rolling contract this week and welcomed back Johnson to make his first start in three months after to a groin injury.
The 62-year-old made four other changes to the side that lost to Chelsea last week while Brede Hangeland and captain Danny Murphy did not make the matchday squad.
Opposite number Danny Wilson faced the tough task of turning over a side 41 places above them in the league standings.
To help his cause, 6,500 travelling Swindon fans made the trip to London to support the League One side on a sunny day at Craven Cottage.
He brought in striker Daniel Ward to join Billy Paynter up front as the Robins looked to reach the fourth round of the competition for the first time since 1996.
Fulham started the brighter of the two sides and took only two minutes to create the first chance when the Robins' defence backed off the advancing Bjorn Helge Riise before he laid the ball off to the in-form Zamora.
The former Spurs striker paused and dummied a challenge but could only shoot wide of David Lucas' goal from 10 yards when he should have hit the target.
Zamora soon made amends for the miss however when he gave Fulham the lead after 16 minutes.
The striker latched onto Stephen Kelly's long ball and chose to stand his ground despite strong pressure from Swindon captain Gordon Greer before calmly lifting the ball over Lucas.
Swindon almost restored parity soon after but Schwarzer saved from Paynter's shot from the edge of the area.
Johnson was clearly ready to make an impression after such a long lay off and he almost doubled Fulham's lead in the 30th minute when he came inches away from turning in Paul Konchesky's cross.
Swindon's physical presence up front was causing problems for Fulham's defence.
Target man Paynter was the main source of attack for Swindon and he calmly controlled Kevin Amankwaah's cross before turning and shooting just over the bar in the 38th minute.
Paynter then shrugged off a challenge from Aaron Hughes, captaining the Fulham side in the absence of Murphy, and laid the ball off to Jonathan Douglas but he missed the target.
Zamora and Johnson, working well in tandem up front, both had chances to double Fulham's lead just before the break but the home side went in 1-0 up after 45 minutes.
Swindon looked to catch the home side off guard early in the second half and Ben Hutchinson should have done better when he curled a tame shot at Mark Schwarzer from inside the box.
Paynter was once again proving a menace up front for Swindon and he won a free kick on the edge of box from which Alan Sheehan tested Schwarzer.
Fulham broke once more shortly after and earned a penalty when Johnson was brought down in the box by the advancing Lucas, who was lucky to receive only a booking for the foul.
Johnson picked himself up but his tame penalty was easily saved by the former Sheffield Wednesday stopper to his right and Swindon were let off the hook.
Lucas was called into action shortly after when he came out to foil Dempsey after a neat turn on the edge of the box.
Johnson pulled away from his marker with 13 minutes left but his fierce shot sailed just wide of Lucas' goal.
That was the striker's last meaningful contribution as he left the pitch to a standing ovation after being replaced by Zoltan Gera.
Wilson threw on youngster Lloyd Macklin in a bid to add life into his attack but Fulham's defensive pair of Chris Smalling and Hughes thwarted any attacks from the Robins.
Paynter continued to press but he could not find a way through and it was Hodgson's side that went into the hat for the fourth-round draw.