Going back even further we were promoted in 1953 and 1961 with a couple of games to spare and relegated in 1949 and 1956 (I believe) having failed to get the right results in our last game of those seasons.
United were already down in 1956 by the last game.
In 1949 United were relegated after drawing with Newcastle. A win would've kept us up.
In fact, I think the last time we had a vital last game of the season where we successfully got the result we needed and it turned out to be vital was the 6-1 win over Spurs at the end of the 38-39 season which saw us promoted (had we lost Wednesday would have gone up instead of us).
We had to win our last game in 1950 to have a chance of promotion; we won 5-0. This put us a point ahead of Wednesday, who had a game left, against an already promoted Spurs. Wednesday drew 0-0 and went up on goal average. A 1-1 draw would have seen the two Sheffield clubs in a dead heat.
It's also usually forgotten about 1988 that United could have gone straight down automatically rather than been in the playoffs had they lost their last game and Reading won; as it turned out, we won at Huddersfield and Reading lost, though we got dumped out of the playoffs anyway.
Oh, and we also lost our last game in 1938 and missed promotion when a draw would have done.
THe only time United have ever escaped relegation on the last day of the season when they could have gone down was 1930, when they won 5-1 at Old Trafford to stay up.