The Red Devils had taken a 2-0 lead via goals from Casemiro and Matheus Cunha. Fulham thought they had pulled a goal back but saw it ruled out after a lengthy VAR check for offside.
However, the visitors produced a late rally with Raul Jimenez scoring from the penalty spot and then Kevin rifling home in stoppage time.
United were not to be denied, as Bruno Fernandes turned his marker and found Benjamin Sesko in the box and the striker hammered an effort into the top corner in the 94th minute to spark jubilant celebrations.
Speaking ahead of the victory, which keeps United in fourth, Carrick left the door open for some late transfer deals.
"Never say never," Carrick said. "A lot can happen and situations can be thrown up.
"Sometimes it is in our control, sometimes not. I am pretty calm. We have shown what we are capable of as a team."
United have been linked with some late moves, and we have taken look at how their new starting XI might look if the rumours come to fruition. Let’s take a look...
Signed late in the summer window, the goalkeeper has proven to be a smart move.
1. GK - Senne Lammens
Signed late in the summer window, the goalkeeper has proven to be a smart move. | Getty Images
Has started at right-back under Carrick in recent games.
2. Diogo Dalot
Has started at right-back under Carrick in recent games. | Getty Images
Is out of contract this summer but has proven a reliable option in Carrick's early days as interim boss.
3. CB - Harry Maguire
Is out of contract this summer but has proven a reliable option in Carrick's early days as interim boss. | Getty Images
Back playing consistently again after fully recovering from an ACL he sustained last February. Has delivered some great performances in recent weeks.
4. CB - Lisandro Martinez
Back playing consistently again after fully recovering from an ACL he sustained last February. Has delivered some great performances in recent weeks. | Getty Images