The 40-year-old joins from Norwegian club Molde and succeeds Malky Mackay, who was sacked on December 27 after his relationship with club owner Vincent Tan broke down.
Wayne Rooney sat out United's 1-0 victory at Norwich City in the Premier League on Saturday, but played for the entirety of the 2-1 home defeat by Tottenham Hotspur on New Year's Day.
Goals from Fernandinho, Yaya Toure and Aleksandar Kolarov in a 3-2 win at Swansea City on Wednesday earned Pellegrini's side another valuable three points in their pursuit of the title.
It is nine years since Arsenal last won a trophy and 10 years since they won the league, but the last-gasp 2-0 New Year's Day win over Cardiff City confirmed their credentials as championship contenders.
The Manchester United manager was incensed by a late decision not to award United a penalty kick in a
Brazil midfielder Oscar had only been on the pitch for a matter of seconds in Chelsea's 3-0 win over Southampton when he went into the book for diving in a failed attempt to win a penalty.
Emmanuel Adebayor and Christian Eriksen scored fpr Spurs to all but end United's hopes of retaining the Premier League title.
Goals from Theo Walcott and Niklas Bendtner gave Arsenal a 2-0 win over Cardiff City to ensure they stayed at the top of the table.
Ole Gunnar Solksjaer, who is coaching Molde in Norway, is expected to replace the sacked Malky Mackay and could be confirmed on Thursday.
Yaya Toure restored City's lead in the 58th with an angled shot into the far corner and Kolarov rifled home in the 66th to secure the points.