The fourth-ranked Russian, who won the event in 2005 and '06, broke Eitzinger's serve five times.
"The first two rounds of a tournament are always hard, so I am glad I came through this one so easily,'' Davydenko said. "I tried to minimize my errors and play as consistently as possible. For me, this event is an important warmup for the French Open.''
Sixth-seeded Andreas Seppi of Italy rallied to beat Ernests Gulbis of Latvia 1-6, 6-4, 6-3. Seppi, a semifinalist in Hamburg last week, struggled with his serve and was broken twice in the opening set.
Also, Jiri Vanek of the Czech Republic beat Dutch wild card Martin Verkerk 7-5, 4-6, 6-2. It was Verkerk's first singles match on the ATP Tour since the 2007 French Open due to various injuries.
Vanek will next play either defending champion Juan Monaco or Carlos Berlocq.
Jurgen Melzer of Austria used six of 16 break points to beat 18-year-old American Donald Young 7-5, 6-2, Ivan Navarro of Spain beat Vincent Spadea of the United States 6-4, 6-4, and Daniel Gimeno-Traver of Spain defeated Werner Eschauer of Austria 6-4, 6-2.
Two other first-round matches were suspended due to rain and will be finished Tuesday. Oliver Marach of Austria led Sam Querrey of the United States 6-3, 3-5, and Dudi Sela of Israel led Mischa Zverev 6-5.
The match between seventh-seeded Mario Ancic of Croatia and Robby Ginepri of the United States had not yet started and was also rescheduled for Tuesday.
Davydenko routs Ljubicic at Hamburg Masters
