The Sky Blues took a second-half lead through Joachim but Johnson popped up twice with identical efforts to stun the home side before a late Joachim equaliser earned City a deserved draw.
The first half saw Gillingham enjoy the lion's share of possession even though Coventry carved out the most meaningful chances.
Most wasteful was City striker Patrick Suffo, who latched onto an inch-perfect cross from Stephen Warnock on 28 minutes but failed to test the keeper.
Paul Smith, who was pulling the strings at the heart of Gillingham's midfield, sent a rasping volley across Gavin Ward's goal on 34 minutes.
Coventry were a much improved side in the second half and nearly broke the deadlock twice before they finally opened the scoring.
First Michael Doyle blazed over on 46 minutes after Suffo laid it expertly into his path on the edge of the box.
Then Warnock found the upper tier of Coventry's West Terrace after the keeper made a hash of claiming a Graham Barrett cross.
The opening goal came from a smooth passing move from Coventry - Steve Staunton found space on the left, but pulled the ball back for Barrett, who in turn picked out Craig Pead's darting run down the right flank.
The full-back's low cross fell perfectly for Joachim to convert first time in off the crossbar.
However, minutes later Johnson grabbed a barely-deserved equaliser when Gillingham broke forward from a Coventry corner.
A long ball forward from keeper Jason Brown found its way to Johnson who caught the defence napping and slotted the ball into the bottom corner from the edge of the box.
Johnson put the Gills ahead with five minutes later after a replica of his first goal but Joachim had the final say when he hooked in his second from the edge of the six-yard box.