Both sides were knocked out at this stage last season, with Sunderland losing 2-0 to Sheffield Wednesday, while Crewe were thrashed 4-0 by Grimsby Town.
Match preview
Indeed, the League Cup has eluded Sunderland throughout their illustrious history, with the Black Cats earning runners-up medals in 1985 and 2014.
With Tony Mowbray's side hoping to end their six-year absence from the Premier League after surprisingly reaching the Championship play-offs in their first season back in the second tier, the former Ipswich Town defender is unlikely to be too concerned about his side's progress in the domestic cups between now and May.
That will ring especially true after losing to his former club 2-1 on Sunday, with goals from Nathan Broadhead and George Hirst either side of half-time giving Sunderland a mountain to climb, especially after Trai Hume was given his marching orders for a second bookable offence 18 minutes before the end of normal time.
However, Mowbray would have been buoyed by his side's positive reaction to that setback, with Dan Neil pulling one back to set up a frantic finish. Should his side carry that spirit into their cup tie on Tuesday, they will almost certainly find themselves in the hat for the second round draw.
Crewe, meanwhile, never came close to bouncing back to League One at the first attempt last season following their relegation, but equally they were never involved in any relegation danger.
However, after losing their top goalscorer Dan Agyei to last season's League Two champions Leyton Orient during the summer transfer window, there are some fears that Crewe could fall out of the Football League for the first time in their history.
Lee Bell, who began his playing career with the Railwaymen before re-joining them in 2010, is embarking on his first managerial role following his appointment last December. He would have been content with drawing 2-2 against promotion hopefuls Mansfield Town in their season opener on Saturday, although Nigel Clough's side did play with one fewer man for more than half an hour during the second half.
Having only made it beyond the first round of the League Cup once in the last six seasons, Bell and his players will be hopeful of improving that record when they visit the North East in midweek, but ultimately their main focus will be on ensuring survival between now and the end of the campaign.
Sunderland EFL Cup form:
Sunderland form (all competitions):
W
W
W
W
D
L
Crewe Alexandra EFL Cup form:
Crewe Alexandra form (all competitions):
L
L
W
W
D
D
Team News
Sunderland are expected to remain without Elliot Embleton and Ross Stewart due to injury for the visit of Crewe on Tuesday.
Mowbray will almost certainly make changes en masse given that this cup tie will kick-off less than 48 hours after their match against Ipswich finished on Sunday, with the likes of Alex Pritchard and Abdoullah Bah expected to be handed an opportunity to impress their manager from the start.
Likewise, Crewe boss Bell may opt to hand several fringe players an opportunity to impress him at the Stadium of Light, with Courtney Baker-Richardson likely to lead the line for them.
Matus Holicek is also expected to come into the visitors' midfield after the 18-year-old was used from the bench during his side's draw against Mansfield at the weekend.
Sunderland possible starting lineup:
Bishop; Huggins, Batth, Triantis, Gooch; Matete, Ba, Evans; Rigg, Semedo, Pritchard
Crewe Alexandra possible starting lineup:
Booth; Cooney, Sass-Davies, Williams, Adebisi; Holicek, Colkett, Griffiths, Long; Baker-Richardson
We say: Sunderland 2-1 Crewe Alexandra
Even with their ludicrously short turnaround between fixtures, Sunderland should still possess too much quality for their opponents on Tuesday.
The Black Cats have aspirations of reaching the Premier League by the end of the campaign, whereas Crewe are very much focused on remaining in the Football League, showcasing the contrast in expectations at both clubs.