The Red Devils conceded early before taking a 2-1 lead by the 72nd minute thanks to an Anthony Martial double.
However, a pair of late headers from Michail Antonio and Winston Reid denied United the chance to move into the top four and they will now rely on results going their way on the final day of the season if they are to clinch a Champions League qualifying spot.
The Red Devils are two points behind neighbours Manchester City, who occupy fourth, and meet Swansea City away on the weekend. United are at home against Bournemouth and require a victory, with City losing in south Wales, to edge above their crosstown rivals.
Of that prospect, van Gaal said: "No it (a top-four finish) is still not gone.
"In the last match of the season we can pass over Manchester City, they can lose and we can win - then we are fourth, you can count I think?
"I don't think it is not likely, in this league it happens a lot and Swansea City has a very good run.
"We have to beat Bournemouth and that is tough.
"We have less points than last year and last year we were qualified two matches before the end and now we are still fighting on the last day. It is less but still we can finish as fourth, we have to think only of that.
"It is a mental blow for the players but against Bournemouth we have to give everything so that we don't give Manchester City a free trip into the Champions League."