Eriksen had a great chance to double his side's lead in first-half stoppage time when he was picked out in the box by Fernandes, but this time he stabbed his shot wide of the near post.
Carlos Vinicius almost put Fulham level just after the break, twisting to fire a shot that goalkeeper David de Gea steered behind, and the Spaniard pulled off another tremendous save to keep out Tim Ream's headed effort from the subsequent corner.
The home side equalised in the 61st minute after a fast-break found midfielder Tom Cairney on the wing whose first-time cross was met by forward Daniel James, scoring first goal for Fulham against his former club -- just one minute after being subbed on.
After coming on in the 72nd minute, the 18-year-old had some promising runs down the left before charging into the box in the 93rd minute to fire his first Premier League goal and snatch all three points.
"I think in the end we're just happy with the three points. It was a hard fought game on both ends but just happy with the win," goalscorer Eriksen said.
"About time! [On scoring his first goal] I owed a few so I'm happy to be on the scoresheet in the last game before the World Cup. It's been a good week.
"Last week wasn't good with the loss at Villa but it's picked up since then. We had a point to prove in the cup, a win today and now onto the World Cup."
United midfielder Fernandes hailed young substitute Garnacho after his decisive intervention.
"He has a lot of talent and his work-rate, together with the talent, makes a big difference with us from the bench. It's not the first time he makes a big impact for us," he said to Sky Sports.
United will now head into the World Cup break in fifth place on 26 points, three points behind Tottenham. Fulham are in ninth on 19 points.
Information from Reuters contributed to this report.