Monreal's move was registered with the FA just 15 minutes before the closure of the January transfer window after the 26-year-old flew to London for a medical.
Arsenal did not confirm his contract details, but he is believed to have signed a four-and-half-year deal worth £60,000-a-week.
Manager Arsene Wenger said: "We are delighted that Nacho Monreal has agreed to join us. We have been monitoring him for some time now and are really pleased that we’ve been able to agree this move today.
"Monreal is a strong left-sided defender with good experience at both club and international level. He is a technically gifted player, a good crosser in the final third and strong in the air. Monreal will add quality to our squad and of course, to our defensive unit. We all look forward to him playing for us."
The move follows the latest injury to first-choice left-back Kieran Gibbs, who went off with a thigh strain in Wednesday's 2-2 draw against Liverpool.
His understudy Andre Santos has come in for stinging criticism following a series of woeful displays.
Malaga have already signed a replacement - Vitorino Antunes of Portuguese club Pacos Ferreira.
However, Arsenal failed in their pursuit of Athletic Bilbao defender Fernando Amorebieta and Barcelona striker David Villa.
The Gunners are sixth in the Premier League, four points behind the Champions League places.