With the Shrimps trailing to Ryan Jackson’s 73rd minute stunner, Obika preserved parity 11 minutes later with his second goal in five outings.
Both goals were out of keeping with a match played against the backdrop of heavy rain and a playing surface which had started to cut up pre-match.
The Shrimps had made one change from the team beaten at Lincoln City in midweek.
Back came Arthur Gnahoua on the left-hand flank with Dylan Connolly the man to make way.
A scrappy start saw Gills keeper Aaron Chapman scuff a clearance straight to Jonah Ayunga, who was unable to capitalise.
Just before the quarter-hour, Dickson-Peters chested a cross back to Ben Thompson, whose fierce shot flew off target.
Anthony O’Connor blocked a Stuart O’Keefe effort and Conor Masterson’s cross-cum-shot was beaten away by Trevor Carson before Morecambe began to get on top.
Ryan Cooney’s cross was narrowly ahead of Adam Phillips and a one-two between Ayunga and Cole Stockton ended with the former’s effort comfortably held by Chapman.
Their best chance came eight minutes before the break when the ball broke to Gnahoua inside the area but, after working the ball onto his left foot, he struck the top of the bar..
The second half began as the first ended with Morecambe having the better possession but unable to fashion a clear chance.
Shane McLoughlin’s cross was too far ahead of Stockton and Phillips, moments before Ayunga saw a first-time shot blocked.
Stockton miscontrolled Ayunga’s cross as one goal had increasingly looked like settling matters.
However, despite Gillingham sitting back and allowing the Shrimps plenty of possession, it was the visitors who took the lead.
Morecambe lost possession and allowed Gillingham to break with the ball worked from the left-hand side across to Jackson, who gave Carson no chance from 20 yards.
Jackson saw a cross headed over by Vadaine Oliver as Morecambe, who brought on Obika and Aaron Wildig during the second period, laboured in possession.
They were about to bring on Connolly when, on 84 minutes, the ball was played into Obika who held off his man before curling a shot past Chapman.
All of a sudden, the Shrimps woke up with Ayunga seeing a close-range effort blocked.
Chapman made a fine save to turn away another Obika attempt, as after four minutes of added time, honours ended even.
Morecambe: Carson, Cooney, O’Connor, Bedeau, Leigh, McLoughlin, Diagouraga, A Phillips (Wildig 65), Ayunga, Stockton, Gnahoua (Obika 74). Subs not used: Smith, Connolly, McCalmont, Fane, Bennett.
Gillingham: Chapman, McKenzie, Masterson, Ehmer, Tucker, Tutonda (Jackson 62), O’Keefe, Thompson, Lee (D Phillips 77), Dickson-Peters (Kelman 86), Oliver. Subs not used: Dahlberg, Lintott, Maghoma, Chambers.
Referee: Peter Wright.
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