The Spaniard picked up two yellow cards inside the opening 26 minutes, the first for needlessly blocking Darren Fletcher's quick free-kick rapidly followed by the second for a rash challenge on the former United man.
Van Gaal was clearly frustrated with Mata's recklessness and cited the Spaniard's sending off as the key reason for United's defeat, per the Daily Mail's Chris Wheeler:
The first was a stupid yellow card. You know it is a yellow card in advance. When you have a yellow card you have to take care of that. The referee can always give another yellow card then and then it is a red card. The second you know you already have one so you have to take that into account. You damage the team.
The decision was the turning point in the match. It is very difficult to play 10 against 11 for 60 minutes, [and] that is the reason why we lost this game. We did not give many chances away. West Brom scored out of three chances and for us it is a big disappointment.
The loss to the Baggies sees United three points off the Premier League's top four in sixth place.
They lost ground on each of the teams above them—as analysed below by Bleacher Report's Dean Jones—and their chances of claiming a spot in next season's Champions League diminished even further:
Sky Sports pundit Thierry Henry described the result on Super Sunday (h/t Sky Sports) as "a massive setback" to their Champions League ambitions.
Indeed, their best chance of qualifying for Europe's top club competition is likely to win the Europa League, and the first leg of their last-16 clash with Liverpool takes place on Thursday.
Van Gaal's side will need a quick response to Sunday's defeat if they are to claim the initiative at Anfield.
The Dutch manager attempted to excuse Mata slightly for his dismissal, suggesting referee Mike Dean "has to know the characters of the players" because it was the 27-year-old's first red card, per Wheeler.
However, it was clear Mata deserved to be shown a red card, even if he is not an undisciplined player, per the Guardian's Daniel Taylor:
Louis van Gaal, a highly experienced manager, arguing a player should not be given yellow cards if he's a nice bloke. Is that in the rules?— Daniel Taylor (@DTguardian) March 6, 2016
Salomon Rondon netted the only goal of the match on 66 minutes to condemn United to a first defeat in five matches following a run of four wins on the bounce.
The visitors were always at a disadvantage following Mata's ejection, but they showed little in the way of intent and did not deserve to win the game.
Their next Premier League game is not until March 20's Manchester derby against City, a must-win game for United if they are to have any chance of finishing the season in the top four.