Djokovic looked poised for victory when he built a 5-2 lead in the tiebreak but Almagro fought back and when Djokovic netted a tame forehand at 5-5 Almagro seized his chance, clinching the second set with a powerful first serve.
Almagro, who missed the second half of last year after foot surgery and has slipped to 174th in the rankings, continued to worry Djokovic with his power at the start of the third set.
Djokovic kept his nose in front on serve though and broke in the eighth game after winning a long baseline exchange.
The three-times Rome champion completed the job on his second match point when Almagro fired a backhand long.
Former Australian Open champion Stanislas Wawrinka also reached the third round but was stretched before overcoming Argentina's Juan Monaco 4-6 6-3 6-2.
Ninth seed Marin Cilic was the highest-ranked casualty, losing in straight sets to Guillermo Garcia-Lopez, but there was no hiccup for seventh seeded Spaniard David Ferrer who came through 6-4 7-5 against France's Richard Gasquet.
In the women's event, top seed and defending champion Serena Williams crushed Anastasia Pavlyuchenkova 6-1 6-3 in her first match since Petra Kvitova ended the American's 27-match unbeaten run in the semi-finals of the Madrid Open last week.
Williams needed 62 minutes to dismiss the former world number 13 in straight sets, smashing 25 winners and converting five of seven break points in a one-sided encounter.
Maria Sharapova also wasted little energy as she moved into the third round. The Russian, seeded three, was 6-2 3-1 ahead when Australian opponent Jarmila Gajdosova retired.
(Reporting by Martyn Herman; editing by Toby Davis)
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