The Portuguese forward, 29, scored for the 15th and 16th times in Europe this season as Real won 4-0 at the Allianz Arena, 5-0 on aggregate.
He broke the record when he swept home following a swift counter-attack.
He then extended the landmark with a low free-kick in the 89th minute. It means he now has 16 goals in 10 matches this season, eclipsing the previous record of 14 in a season jointly held by Messi (2011-12, Barcelona), Ruud van Nistelrooy (2002-03, Manchester United) and Jose Altafini (1962-63, AC Milan).
The ex-Manchester United forward has now scored 67 goals in his Champions League career, the same number as Messi and four shy of the record which is currently held by former Real and Schalke striker Raul.
