The Argentina international was injured after a heavy collision and tried to play on before being replaced mid-way through the second half at Old Trafford.
Rojo has only just returned to action after a shoulder problem.
However, the Manchester United boss said that he is hopeful that the suspected ankle injury is not too serious, as the Red Devils turn their attentions towards their home clash with Watford on Wednesday night.
Speaking to Sky Sports after Sunday’s game, Van Gaal said: “I don’t think it is serious; it was a kick.
“We have to wait and see, of course. It was also a good position that I gave him the first minutes to play.
“My plan was already 60 minutes. He didn’t reach the 60 minutes because of the kick.”
Sunday’s battling win over Arsenal left United three points behind fourth-placed Manchester City heading into the final 11 games of the season.
United take on Watford on Wednesday, before travelling to West Bromwich Albion in the Premier League on Sunday.
MORE: Man United latest news
MORE: Arsenal latest news
MORE: Chelsea latest news
MORE: Liverpool latest news