Speed, deputising for Kevin Blackwell at the post-match press conference, said: "If we can show those levels of energy and commitment then we will be there or thereabouts at the end of the season.
"I'm sure of that. Palace, especially with Neil being there, are always going to be tough to play against but I thought we stood up to them well.
"We out-passed them and I thought we deserved it. The best team won.
"But we are taking nothing for granted because this division can bite you on the bum if you take your foot off the pedal at any time."
United put the visitors to the sword with goals in either half.
Lee Williamson, netting for the first time for his new club, fired them into a deserved lead when he stroked home following good work by Ched Evans and Stephen Quinn.
Evans, described by Blackwell as 'a young Alan Shearer' on the eve of this fixture, lived up to his billing with a commanding display before being withdrawn during the closing stages.
The Wales international, having seen his shot deflected onto the post by Danny Butterfield during the opening exchanges, then tested Julian Speroni's reactions from a Jamie Ward corner.
Evans served notice of United's intentions soon after the re-start with another good effort while Ward was denied what appeared to be a clear-cut penalty midway through the second half when Clint Hill handled his centre in the box.
United, however, were not to be denied and Quinn finished emphatically at the death to ensure the final scoreline reflected their dominance.
Warnock, who three years ago led United to promotion before taking charge at Selhurst Park, said: "We came here in very difficult circumstances. Five of our lads came down with something - we think it was something they ate.
"They were never really at the races but they gave everything that they could and I can't have a go at them.
"The best team won and now we move on."