The Colombia star, a deadline-day recruit on a season-long loan from Monaco, grabbed the winner as United edged out Everton 2-1 in the Premier League on Sunday.
Falcao converted in the 62nd minute to secure the points after Steven Naismith had cancelled out an opener from another of United's high-profile new boys, Angel Di Maria.
The decisive goal delighted manager Van Gaal, who said: "He [Falcao] needed that. He was forcing himself too much to score. Now he has, it'll be better after the international break."
Falcao's goal came after the 28-year-old had uncharacteristically failed to convert some earlier opportunities.
After the game, he described his first United strike as "amazing", but conceded: "There was a little bit of relief."
Having claimed back-to-back victories for the first time under Van Gaal following a shaky start to the season, United now lie fourth in the Premier League.
They will resume action after the international break with a trip to West Brom.
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