Real hammered Sevilla 4-1 on Sunday night with former United forward Cristiano Ronaldo scoring a hat-trick.
The victory means Real have only lost once in 2013, with Granada inflicting that defeat last weekend.
Madrid coach Jose Mourinho has complained on several occasions that match scheduling has worked against his team but with United facing Everton at 1600 GMT on Sunday, Real have an extra day's rest ahead of Wednesday's last 16 first leg.
"Real Madrid are coming into the match in good form, and not because of (last night's) result," said Karanka.
"Apart from the match against Granada, the team has been showing good things throughout January. (Against Sevilla) we played well and have given a bit of a rest to key players and the match comes at a good time for us."
On the extra day's rest, Karanka said: "We've proven it ourselves - it does show.
"But they (United) have a bit of a cushion (at the top of the Premier League) and I'm sure they'll be making rotations and resting important players."
Ronaldo, who joined Real Madrid from United in 2009, was in ominous form on Sunday, following up Karim Benzema's opener with a 33-minute hat-trick before being substituted in the 63rd minute.
Karanka is delighted with the form of Ronaldo, who has scored 29 goals in 29 games for Real this season, ahead of this Wednesday's game at the Bernabau.
"He has had that admiration (from the fans) for a long time and he knows how important he is for the team," the Madrid number two said. "It is important that a player like Cristiano is at this level."