United edged through to the quarter-finals of the Europa League with a 2-1 aggregate victory over Rostov, Juan Mata scoring the only goal of a tight second leg.
The game came three days after 10-man United lost in the last eight of the FA Cup to Chelsea and ahead of a Premier League trip to Middlesbrough at midday on Sunday.
Mourinho told BT Sport: "I was afraid of extra-time and I knew I had Marcos Rojo in trouble, so it was a difficult game.
"It is difficult to play Monday night with 10 men, it is difficult to play now and it is difficult to play on Sunday at 12 o'clock so we have lots of enemies, but the boys were fantastic.
"We have lots of things going against us and we will have to fight and we will probably lose on Sunday. Fatigue has a price.
"I will remember forever when I spoke to the UEFA delegate in Rostov. He told me if any of our players gets injured, the insurance paid. Whoever decided the Monday and Sunday games probably thinks the same way."
Paul Pogba may well have paid the price for the hectic schedule, with the midfielder sustaining a hamstring injury in the opening moments of the second half.
Mourinho confirmed he will definitely miss Sunday's game as well as France's internationals against Luxembourg and Spain, while Daley Blind also left the pitch early after taking a knock to the head.
"I don't know for sure how long but for sure no Middlesbrough and no national team," Mourinho said of Pogba.
Rojo, meanwhile, was so low on energy levels that he ate a banana during the second half.
"The player asked for a banana," said Mourinho. "But it's not to laugh at all. It's to respect the players."
United should have been ahead after a first half they dominated but Zlatan Ibrahimovic twice hit a post, Rostov goalkeeper Nikita Medvedev saved well from Rojo and Pogba and Henrikh Mkhitaryan missed a one-on-one.
Mata's goal calmed the growing nerves but United still needed three fine saves from Sergio Romero, including from Christian Noboa's free-kick with the final act of the game.
"I think Rostov is a very difficult team to play against because they defend with a block of 10 players," said Mourinho.
"It was difficult for everyone. Only Bayern at home scored early and had an easy job, but Atletico Madrid, Anderlecht, Ajax, Sparta Prague, Man United, it was difficult to play against them.
"Our plan was to kill the game when we had energy for that. We played in the first half like we were the team that needs to score, we tried to kill in the first half, we hit the post, the goalkeeper saved, we missed chances, so they let the game go open and then when we were dead they were trying. My players they fought hard to keep the result and go through."
United will find out their quarter-final opponents at Friday's draw, and it would undoubtedly be a disappointment if they do not now go on to win the competition, and the Champions League place that comes with it.
It appears the best chance of Mourinho's team qualifying for the elite competition next season, particularly given United are the only top-six team left in Europe.
"(They) will be playing one match per week like some of them have been all season," added Mourinho of United's rivals. "They don't know what fatigue is. We fight until the limits."