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The Hampshire part-timers, nicknamed the Spitfires after the World War 2 plane that first flew from the local airfield, were on the verge of causing the day's one genuine shock when they led through Dorian Dervite's own goal.
They came back to earth with a thud when debt-ridden Championship side Bolton levelled three minutes from time through Darren Pratley to set up a replay.
On a day of 25 ties, holders Arsenal, bidding to become the first club to win the trophy three times in succession since Blackburn Rovers in the 19th century, came from a goal down to beat Premier League strugglers Sunderland 3-1.
Toiling Manchester United were minutes away from a drab stalemate with third tier Sheffield United at Old Trafford before Wayne Rooney converted a stoppage-time penalty to give the 11-time winners a 1-0 victory.
Manchester City and Everton also made it through to the fourth round, City winning 3-0 at Norwich City in one of four all top-flight clashes and Everton ending the hopes of fourth- tier Dagenham and Redbridge with a 2-0 win at Goodison Park.
League Two (fourth tier) Wycombe Wanderers made sure their name will be in the hat for the fourth round along with Exeter City after a 1-1 draw at home to the Premier League's bottom club Aston Villa who have now gone 16 matches without a win.
Exeter held Liverpool to a 2-2 draw on Friday.
Hartlepool, also Football League bottom rung, took the lead against former winners Derby County but lost 2-1 while Portsmouth, FA Cup winners in 2008 but now floundering in League Two, earned a 2-2 draw at Championship club Ipswich Town.
Watford beat Premier League rivals Newcastle United 1-0, Crystal Palace overcame Southampton 2-1 and West Ham United edged past Wolverhampton Wanderers 1-0 at Upton Park.
Staff at Eastleigh, who play in the National League, one rung below the Football League, worked overtime to make their boggy pitch playable after heavy rain, and the tie got the go-ahead only after a morning inspection.
Their efforts were rewarded as Eastleigh gave Bolton a fright, taking the lead after 51 minutes when Dervite prodded a low cross into his own goal
With the majority of the 5,000 crowd counting down the minutes, Bolton proved party-poopers with Pratley firing home a loose ball to deny Eastleigh a place in the fourth round for the first time in their 70-year history.
"We deserved to win," Eastleigh manager Chris Todd said.
"We're the National League team, they are the Championship side, three divisions above us, but I heard shouts from their bench saying 'get it in the corner'."
Bolton are rock bottom of the Championship with debts of around 170 million pounds but their late escape showed they still have a "beating heart" according to manager Neil Lennon.
Premier League leaders Arsenal notched a 13th consecutive FA Cup win with goals from Joel Campbell, Aaron Ramsey and Olivier Giroud after Jeremain Lens put Sunderland ahead.
Sergio Aguero scored a superb individual goal and set one up for Kelechi Iheanacho as Manchester City eased past Norwich.
Arouna Kone and Kevin Mirallas earned Everton victory over Dagenham who are 79 places lower in the football pyramid.
No Premier League clubs suffered defeats by lower league opponents, although West Bromwich Albion came close, James Morrison stealing a 2-2 draw at home to Championship side Bristol City in the 95th minute.
Bournemouth won 2-1 at Championship promotion hopefuls Birmingham City while former England striker Peter Crouch was on target as Stoke City won 2-1 at third-tier Doncaster Rovers.