Marouane Fellaini???s volley and James Wilson???s late strike gave Louis van Gaal???s men the points in a game which the hosts were more than competitive.
Newspaper reports in the week suggested Redknapp would lose his job with defeat on Saturday, but the Rs boss slammed the rumours and said he is sure he will keep his side in the Premier League.
He said: ???One person writes a load of rubbish in the newspaper ??? why would I be bothered?
???Do you think I go home at night and don???t sleep over that? You saw the effort today and I can???t ask for any more than that.
???We???ll be OK. QPR won???t go down this year.???
Wilson???s strike in the fourth minute of stoppage time put the gloss on the scoreline for United, but the hosts had some good chances of their own as Charlie Austin was denied by some excellent goalkeeping from David De Gea.
While disappointed with both of the goals, Redknapp was impressed with his side???s work rate and hailed QPR’s home record despite Saturday’s defeat.
???I???m really pleased with the performance,” he added. “Even at 1-0 in the last 10 minutes we???re pushing for an equaliser. The second goal happened as we had left one at the back and they had three attacking us.
???We were bombarding their box trying to get an equaliser and they obviously broke away and scored. But I thought the work rate of the team was first class.
???At half time we came in and I felt we could get something. The goal was disappointing. We switched off and they got in behind us. We didn???t deserve to be behind. We kept going and 2-0 we didn???t deserve. Overall, I???m very pleased with the team.
???We???ve got a great home record and that was as good as we???ve played at home.???