Pogba watched the action from the stands at Old Trafford, but was a hot topic of discussion in Mourinho's post-match interview on Sky Sports.
Mourinho needs to back his star player, and handing him the armband seemed like a positive step.
We haven't played great, we haven't been unlucky and opposition weren't lucky.
Mino Raiola, Pogba's agent, has himself been working on Pogba's exit from the club for months, negotiating personal terms with Barcelona in the later weeks of the summer transfer window.
Championship side Derby nearly beat United within 90 minutes, coming from behind to lead 2-1, but Marouane Fellaini's late equaliser sent the third-round tie to penalties.
Jose Mourinho confirmed Paul Pogba has been stripped of the vice-captaincy at Manchester United, but insists they have not fallen out.
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'The only truth is that I made the decision for Paul not to be the second captain any more, ' said Mourinho.
"Mourinho's message to the United squad, delivered at Carrington today, was created to underline his authority and remind the players that he is in charge", added The Times report.
Pogba still turned up at Old Trafford, watching from a private box, but disappeared before the shoot-out. They played so well against Brentford and I knew how much they all wanted to play.
The East Midlands side appointed untested manager Lampard in May and since the beginning of the 2018-19 season he has led the Rams to sixth in the second-tier Championship with five wins, one draw and three losses.
Pogba rejoined United in 2016 and his arrival was marked by a wide-ranging social media campaign which sources say Mourinho felt put the player above the club.
"As someone who wants Manchester United to do well, I'm fed up of waking up and seeing these headlines".
Before the first Derby County penalty was taken, the United manager spoke to Lampard.