Throughout this season, the veteran has come in for criticism from several quarters after a number of below-par performances for United, but he played a key role as Louis van Gaal's men secured a 1-0 victory over rivals Manchester City in the derby at the Etihad Stadium.
While Rashford received the plaudits for his brilliant individual goal, Murphy paid credit to Carrick after he helped his side keep a clean sheet in a crucial game for the club.
He said: "He plays a lot of passes that people don't say "wow" about, but players know how crucial they are to get the team out. His positional play was great and he made a tackle late in the game that was absolutely superb.
"For me, I would take him to the Euros. Whether you start him or not is a different debate but for his experience and calmness, I would definitely have him there."
Carrick has made 33 appearances for Manchester United in all competitions this season, but he has featured on just one occasion in the national team.
He played 89 minutes for the Three Lions during the 2-0 defeat to Spain in November to take him up to 34 caps since making his debut in 2001.