The 29-year-old was a second half substitute as United bowed out of the competition at the round of 16 stage with a 2-1 aggregate defeat to the Rojiblancos.
Pogba's two kids were at home when the incident occurred but his wife was also in attendance at the match.
"Last night our family's worst nightmare was realised when our home was broken into and burglarised while our babies were sleeping in their bedroom," Pogba wrote in a statement published on the United man's Twitter account.
"The burglars were in our home for less than five minutes but in that time they took from us something more valuable than anything we had in our home... our sense of safety and security.
"This occurred during the final minutes of last night's match match when they knew I wouldn't be home. My wife and I rushed home not knowing if our children were safe and unharmed. As a father there is no feeling worse in this world than not being there to protect your children and I sincerely hope that no one ever has to feel what I felt last night."
An injury-hit campaign has seen Pogba make just 23 appearances for Manchester United across all competitions this season, although he has still contributed one goal and nine assists.
Pogba's contract at Old Trafford expires this summer and there are no indications that he wants to extend his stay at the club.
