Dementieva failed to win three break points in a 14-minute opening game and she had to fight off two break points on her serve before holding for 2-2.
But Kirilenko faltered in making a forehand error to give Dementieva a 3-2 lead, and after holding serve at love she broke again when her opponent double-faulted on break point.
Despite double-faulting twice on set point Dementieva served out the set, and then broke to lead 1-0 in the second set when Kirilenko made another forehand error.
Dementieva all but sealed her victory by breaking again for 5-2 with a forehand winner.
"It was not that easy and I think it was a good match for the first match on (a) grass court,'' Dementieva said. "There's such a short time after the French Open, but I think it's quite interesting and it's always fun to play on grass.''
Nadia Petrova, runner-up last year to Agnieszka Radwanska, won the last five games to defeat Serbia's Ana Ivanovic 6-1, 4-6, 6-4.
After losing the first four games of the match, Ivanovic came back into the match in the second set and broke to lead 5-4.
She also broke twice in the final set to build a 4-1 lead, before the seventh-seeded Russian rallied.
"It has been a difficult first-round match but I had fun out there,'' Petrova said. "I wanted to play my best tennis and leave everything I could out there and I was very happy with my performance after 4-1.''
For the first time, an ATP event is taking place alongside the women's traditional Wimbledon warmup event.
Fourth-seeded Fabrice Santoro of France beat American Robert Kendrick, and eighth-seeded Guillermo Garcia-Lopez of Spain overcame Russia's Evgeny Korolev 6-2, 6-7, 6-2.
Fire break through against Reds
Reds eyeing crucial three points against Revs
Petrova holds off Bondarenko at Ponte Vedra
