Not since the 1880s has a team lifted the world's oldest trophy three years in a row, but the Gunners will be confident of taking a step closer on Saturday when third in the Premier League takes on third in the Championship.
The FA Cup won't be top of Arsenal's priority list this season, but it is certainly a competition with which they have a great affinity.
They have lifted the trophy more often than any other side (12), including each of the last two years, and are bidding for a historic treble this season.
The fact that this was also the trophy that ended Arsenal's nine-year drought must also endear it further to Wenger, whose qualities as boss were severely questioned during that bleak spell.
Many fans will not be entirely satisfied until they end their wait for a league title, however, which is why, despite the history Arsenal have with the FA Cup, it must play second fiddle once again this term.
Questions over the mentality of the Arsenal squad have resurfaced in recent weeks following a three-match winless streak that continued on Sunday with defeat to London rivals Chelsea.
For the second time this season, Chelsea were aided by a red card against the Gunners when Per Mertesacker was sent off after just 18 minutes, and Diego Costa's strike five minutes later proved to be the difference at the Emirates.
It was only Arsenal's second defeat in 13 matches, but saw them lose ground in the title race, dropping below Manchester City and three points adrift of Leicester City.
They have not won since a 3-1 victory over Sunderland in the third round of this competition - their 13th straight FA Cup victory and their seventh in a row at home. The last team to beat Arsenal in the FA Cup were Burnley's Lancashire rivals Blackburn Rovers back in 2013.
Their home form in all competitions has been highly impressive too - the defeat to Chelsea was their first at the Emirates Stadium in their last 11 outings there and only the second time that they had failed to pick up maximum points in that run.
Recent form: W
Recent form (all competitions): WWWDDL
Burnley could also be forgiven for having their minds elsewhere heading into this tie as they are embroiled in a similar top-of-the-table tussle to their opponents.
While the Premier League title is on the Gunners' agenda, it is promotion back to the top flight that Burnley are chasing, and the recent signs are looking promising.
Four points is still the gap to the automatic promotion places, but Burnley are unbeaten since Boxing Day and have won their last four outings in all competitions.
The formula that won promotion for the Clarets last time has since been altered, though. They had the best defence in the league and scored 12 fewer goals than the highest scorers in 2013-14, but this time around they boast the league's best attack, while five teams have conceded either fewer or the same number as Burnley.
It is the attacking aspect that has come to the fore in recent weeks, with their six-match unbeaten streak seeing them score 16 goals, including 12 in their last three outings.
Perhaps the biggest statement of intent so far came on Monday night when fellow promotion hopefuls Derby County came to Turf Moor, only to be sent packing on the wrong end of a 4-1 defeat.
That match was the beginning of a huge four-game streak for the Clarets which, in addition to Arsenal, sees them take on sixth-placed Sheffield Wednesday and Championship leaders Hull City in the space of a week.
Things have improved away from home too, with Burnley having won their last three matches on the road in all competitions - the same number of victories that they had managed in their opening 13 away games of the season.
However, their FA Cup record is not quite as impressive. The victory over Middlesbrough in the third round was the first time that they have won in this competition since 2011, which was also the last time that they reached the fifth round.
They have not beaten a top-flight team in the FA Cup since 2009 - a season that saw them knocked out by Arsenal at the Emirates in the very next round.
Recent form: W
Recent form (all competitions): WDWWWW
Arsenal will be without Mertesacker for this match following his red card against Chelsea at the weekend, which could see Gabriel Paulista drafted in.
Burnley have fewer options when it comes to rotation, and Dyche could pick an unchanged side to the one that dispatched of Derby on Monday night.
Arsenal possible starting lineup:
Ospina; Bellerin, Gabriel, Koscielny, Gibbs; Elneny, Chambers; Oxlade-Chamberlain, Sanchez, Campbell; Walcott
Burnley possible starting lineup:
Heaton; Lowton, Keane, Mee, Ward; Boyd, Jones, Barton, Arfield; Vokes, Gray
Head To Head
Arsenal have won four and lost none of the last five meetings between these two sides in all competitions, with Burnley's most recent victory coming in the League Cup in 2008.
They have met seven times in FA Cup history, with Arsenal winning each of the last five. You have to go back to 1898 for Burnley's last victory over the Gunners in this competition.
Their most recent meetings came in the Premier League last season, when Arsenal won 3-0 at the Emirates and 1-0 at Turf Moor.
The Dugout Podcast
Hit play below to listen to Sports Mole's new podcast 'The Dugout' for our take on the week's biggest football talking points:
We say: Arsenal 2-0 Burnley
Burnley will be no pushovers and come into this match in fine form, but this is a step up from anything they have faced this season. Arsenal at the Emirates is a huge test for any side, and the Gunners should have enough to avoid an upset and progress to the fifth round.