The Belgian ultimately scored the winner in the 91st-minute for the Reds, heading home after coming on as a second-half substitute for Ander Herrera.
He had been involved in a chance just minutes earlier when his header struck back off the post and Marcus Rashford tapped home, only for it to be ruled out for offside.
Mourinho said Fellaini came on for Manchester United to exploit a change from Arsenal in terms of their defensive block.
“I knew they were going to drop the block,” Mourinho said. “I felt we needed a second presence.
“He scored a beautiful goal.”
The midfielder rose above three players marking him and directed his header into the corner.
“It’s a goal that mathematically qualifies us for the Champions League,” Mourinho added.
Fellaini has changed a game for Man United once again. For all his faults, his effectiveness is undoubted. He had a key role in the UEFA Europa League run last season, when United won the tournament, and has been key in EFL Cup and FA Cup victories in the last two seasons.
His contract expires in the summer but after months of assumptions that the 30-year-old would reject a new deal at the club in favour of a new challenge, Mourinho indicated on Sunday evening that Fellaini looks set to put pen to paper on a new deal. He celebrated by grabbing the United crest on his shirt.