Almagro defeated fellow Spaniard Marcel Granollers 6-3, 6-4 on the outdoor red clay, after Argentina's Monaco outlasted Frenchman Jeremy Chardy in a long second-set tie-breaker to win 6-0, 7-6 (13/11).
The rain held off at the Real Club de Tenis until former No 1 Juan Carlos Ferrero was honoured after Almagro's match for his career that ended with his retirement at age 32 last summer.
Seven-times winner Rafael Nadal is scheduled to play Benoit Paire of France later, while Tomas Berdych and Milos Raonic will also be in action.
The fourth-seeded Almagro used his reliable ground game to ease past Granollers, converting three of four break chances. Granollers, meanwhile, managed only one break from seven opportunities.
Almagro unleashed a well-placed backhand shot down the line to take a one-set lead, and claimed the second set after saving a break with three straight points in the clincher.
He said he had recovered from a drop in blood sugar that had caused him to struggle through the second round.
"Physically, I feel much better than yesterday," said Almagro. "(Granollers) is always a tough opponent. We know each other so well from the circuit and the Davis Cup team. It's too bad we had to meet."
Also, Dmitry Tursunov, the Russian qualifier who upset top-seeded David Ferrer on Wednesday, was beaten by Thomaz Bellucci of Brazil 4-6, 6-1, 6-3.
Results on the fourth day of the Barcelona Open here on Thursday (x denotes seeded player):
Third round
Juan Monaco (ARG x7) bt Jeremy Chardy (FRA x10) 6-0, 7-6 (13/11)
Thomaz Bellucci (BRA x16) bt Dmitry Tursunov (RUS) 4-6, 6-1, 6-3
Nicolas Almagro (ESP x4) bt Marcel Granollers (ESP x15) 6-3, 6-4
Martin Klizan (SLO x11) leads Philipp Kohlschreiber (GER x8) 6-1, 6-7 (2/7), 4-3, rain