It was the Second Division club who took the lead on nine minutes when Scott McGlashan made it 1-0 to The Red Lichties.
However, the Edinburgh non-leaguers drew level when Keith McLeod made it 1-1 and Craig Manson put them ahead just before the break.
Phil Johnson made it 3-1 to the visitors before Manson scored his second, as Spartans went on to rout the league hosts.
Spartans' reward for making round four will be a home derby tie against Scottish Premier League side Livingston.
Old Firm rivals Celtic were made to work hard for their third round win by First Division side Ross County but went through 2-0 at Parkhead.
Two goals in five second half minutes from John Hartson and Paul Lambert was enough to see off a spirited effort from the Dingwall club.
In Saturday's early match, Scottish Cup holders Rangers advanced to the next round with a comfortable 2-0 away win at Hibernian.
Alex McLeish's set aside their woeful display at Old Firm rivals Celtic to put in a solid third round performance at Easter Road, as goals by Shota Arveladze and Peter Lovenkrands sent the Ibrox club through.
An scrappy first half was turned on its head ten minutes before the break, when The Gers made the most of the game's first real chance on the break.
Lovenkrands was able to get down the left flank before crossing into the Hibs box to find substitute Hamed Namouchi at the far post.
His header back across goal found an unmarked Arveladze, whose diving header gave Rangers a 1-0 lead.
Alex McLeish's side would double their lead four minutes after half time when Lovenkrands made another darting run into the Hibees box before finishing near post past goalkeeper Daniel Andersson and celebrating with a somersault.
The second goal ended the match as a contest and the game ended with Rangers a step closer to retaining the trophy.
Aberdeen and Dundee will have to replay at Dens Park following a 0-0 draw at Pittodrie.
It was a good result for The Dark Blues, who played most of the second half with ten men after Mark Fotheringham was sent off for a second bookable offence.
Goals from Jason Lee and Canadian international Davide Xausa was enough for Falkirk to see off First Division rivals Ayr United 2-1. David Craig replied for The Honest Men.
Clyde defeated Third Division Gretna 3-0 with two goals from Andy Smith and a single strike by Jack Ross.
Dunfermline Athletic won the all-SPL tie with Dundee United 3-1 at East End Park.
The Pars grabbed two goals from Stevie Crawford and a third goal from Craig Brewster while Derek McInnes scored a consolation for The Terrors.
Sean O'Connor's 75th minute strike gave Queen of the South a 1-0 victory over Second Division club East Fife while a brace from Brian McPhee gave Hamilton Academical a 2-0 win over Cowdenbeath.
SPL side Hearts eased past Second Division side Berwick Rangers 2-0, with a first half own-goal by Mark Cowan and a second half strike from Joe Hamill.
Inverness Caledonian Thistle thrashed First Division rivals Brechin City 5-1, with Paul Ritchie scoring a hat-trick for the home side.
David Bingham and Rob McBain added to Caley Thistle's total while Charlie King was able to reply for The City.
A Davide Fernandez strike was enough to give Livingston a 1-0 win over Third Division side Montrose at Almondvale, with the prospect of facing Cup heroes Spartans to come in the next round.
SPL strugglers Partick Thistle saw off Second Division side Morton with a 3-0 win at Cappielow Park.
David Rowson and two penalties from James Grady saw the Glasgow club through, as the home side finished the match with ten men after Scott Bannerman was sent off in the dying moments for a professional foul.
Kilmarnock made no slip ups at First Division side Raith Rovers following a 3-1 victory at Stark's Park.
Gary McSwegan, Gary McDonald and Colin Nish gave Killie a 3-0 half time lead before Vincent Talio replied after the break for the home side.
Motherwell defeated First Division side St Johnstone 3-0 at McDiarmid Park with two goals from David Clarkson and a third goal from Scott McDonald.
Finally, goals by Barry Laverty and David McKenna gave First Division side St Mirren a 2-0 win over Second Division opponents Airdrie United.
Click below for match summaries:
Scottish FA Cup
Aberdeen 0-0 Dundee
Arbroath 1-4 Spartans
Ayr United 1-2 Falkirk
Celtic 2-0 Ross County
Clyde 3-0 Gretna
Dunfermline Athletic 3-1 Dundee United
East Fife 0-1 Queen of the South
Hamilton Academical 2-0 Cowdenbeath
Hearts 2-0 Berwick Rangers
Hibernian 0-2 Rangers
Inverness CT 5-1 Brechin City
Livingston 1-0 Montrose
Morton 0-3 Partick Thistle
Raith Rovers 1-3 Kilmarnock
St Johnstone 0-3 Motherwell
St Mirren 2-0 Airdrie United
