Aberdeen v Rangers will kick off at noon on Sunday 28 October.
Hearts v Celtic will then follow at 7.45pm.
The Scottish Professional Football League made the announcement following a meeting on Thursday.
Both League Cup semi-finals normally take place at Hampden, one on Saturday and the other on Sunday, but both Glasgow clubs are in Europa League action on the preceding Thursday night.
Celtic face RB Leipzig in Germany while Rangers will host Spartak Moscow and Ibrox manager Steven Gerrard said it would be “unfair” to ask either of the Glasgow teams to play again after just one day’s rest.
The SPFL’s solution is to playing both ties on the Sunday.
Murrayfield had been suggested as possible alternative but the SPFL says it is contractually obliged to play both semi-finals at football’s national stadium.
SPFL secretary Iain Blair said: “We’ve been working hard with Police Scotland and Hampden Park on contingency planning for exactly this scenario and, together, we examined every possible permutation.
“The crowded fixture calendar means that this is the best and most practical solution as any alternative had the knock-on effect of causing major fixture congestion, not only for Celtic and Rangers, but for several other SPFL clubs.
“We met with representatives of all four clubs earlier today and explained that this was the only feasible outcome.
“The fact that both Celtic and Rangers are in Europe, and will hopefully be playing in the Europa League after the New Year, is obviously good news for Scottish football, but it does mean that our capacity for rescheduling games is significantly curtailed.
“We can confirm that Police Scotland have advised that the Aberdeen vs Rangers game should be played first.”
A Hampden Park Limited spokesman said “We have already started the detailed planning process which is standard practice for all major events at the Stadium. Hampden is the home of Scottish football and we all look forward to assisting our SPFL colleagues with the delivery of two very exciting Betfred Cup semi-finals at the Stadium.”