Apart from the first 20 minutes the Glovers were outplayed by the non-League side and could have no complaints with the outcome.
Jamie McGuire steadied the visitors` nerves with a well-taken goal after 28 minutes. Peter Till did well on the right and when he crossed towards the penalty spot, McGuire threaded a low shot past Gareth Stewart.
Yeovil's anxiety increased, and following a spate of bookings, defender Bondz N'Gala was sent off for a second yellow card.
Fleetwood also had a man sent off when skipper Steve McNulty was given his marching orders, also for two yellows, in the 85th minute.
That did not stop them from adding a second goal in stoppage time, though, as Jamie Vardy beat a defender before slipping the ball home.
The Shrimpers paid the price for Bilel Mohsni's first-half red card as the home side went on to dominate with a composed display.
Oldham goalkeeper Alex Cisak had to make two fingertip saves in the first 18 minutes, denying both Neil Harris and Mohsni from distance.
Latics defender Zander Diamond went close, but the League Two visitors threatened again as Cisak's agility kept out Harris.
Southend had Mohsni sent off in the 38th minute, the Frenchman dismissed for two yellow cards earned by late challenges on Kieran Lee and Dean Furman in the space of 15 minutes.
Oldham took control and grabbed a 58th-minute breakthrough when James Wesolowski's lay-off set up Taylor to smash home from 10 yards.
Diamond hit a post as Paul Dickov's side chased a second, but Southend's 10 men created little in return.
Conference South side Salisbury produced a thrilling extra-time FA Cup shock at Grimsby as a 3-2 victory set up a third-round tie away at Sheffield United.
A penalty seven minutes from the end of added time from Stuart Anderson sealed a famous and deserved win at Blundell Park.
The Wiltshire side had the better of the chances in the first half. Right-back Ben Adelsbury hammered a 25-yard effort from a short free-kick straight at Grimsby defender Charlie I'Anson who looked suspiciously behind the goalline but kept it out in the ninth minute.
Then, in-form Grimsby goalkeeper James McKeown had to be at full-stretch to palm away a Brian Dutton header, while City`s FA Trophy hat-trick hero Dan Fitchett somehow nodded onto the top of the crossbar from just a few feet a minute later.
Salisbury kept pressing but could have gone behind before the break when Grimsby hit the woodwork themselves and questioned whether Anthony Elding's header bounced behind the line.
Thirty seconds after half-time, though, Fitchett had his moment when his curler beat McKeown in front of the travelling Whites fans.
It looked to be the winner until Grimsby substitute Rob Duffy nodded in a leveller in the second of four added minutes to take it into extra-time.
Then he placed home 30 seconds after the restart to put the home side ahead.
But Dutton stooped to head home from an acute angle to make it 2-2, before Anderson became the hero from the penalty spot when Shaun Pearson hauled down Fitchett in the box.
Dagenham earned a third-round FA Cup tie at home to Millwall after winning a penalty shootout 3-2 following a drab goalless draw with 10-man Walsall at the Banks's Stadium
To the delight of their fans, spot-kicks by Michael Spillane, Medy Elito and Brian Woodall were successful, while Jon Macken and George Bowerman succeeded for Walsall but Alex Nicholls and Claude Gnakpa missed.
The Daggers deserved to go through as they were the better side, though the Saddlers were handicapped when they lost central defender Manny Smith who was stretchered off with a head injury 10 minutes from the end of normal time.
Walsall goalkeeper James Walker was named man-of-the-match, which indicated the Daggers` superiority, and he twice saved well from Abu Ogogo and Woodall in the first seven minutes.
Daggers goalkeeper Chris Lewington made his best save from Jon Macken after 79 minutes.