The Red Devils were heading towards a frustrating home draw after former player Henrikh Mkhitaryan struck to cancel out Paul Pogba's stabbed opener.
Fellaini beat three Arsenal defenders to head past David Ospina in the 92nd minute of the match, however, earning a 2-1 win for his side which Mourinho believes was well deserved overall.
"I think if it was a draw it wouldn't be unfair. Arsenal did a good game, they defended well, but we were the team that tried to win that created a few more chances," he told Sky Sports News. "But that is normal.
"We were a bit lucky to score, but we were stronger in the last 20 minutes. We won three points that kill the Champions League story. But we still need four points to finish second which is what we want."
United completed a Premier League double over Arsenal for the first time since the 2011-12 campaign.