The home side bossed the first half and deservedly led at the break, but after manager Peter Jackson made half-time changes, the visitors turned things around courtesy of an own goal and the wonderful Dany N'Guessan strike.
From then on in it was a case of Rochdale bombarding the Lincoln rearguard who finally crumbled with McEvilly's 84th minute goal.
Rochdale took the lead on 27 minutes when following a patient passing move winger Adam Rundle easily beat veteran full-back Frank Sinclair, but saw his powerful shot saved by Rob Burch, only for the livewire Adam Le Fondre to pounce with a close-range finish from the rebound.
The home side were left to rue missed first-half opportunities as within two minutes of the restart Lincoln pulled level when defender Tom Kennedy could do nothing but deflect an Anthony Elding cross into his own goal.
On 62 minutes Frenchman N'Guessan gave the visitors a lead when he fired home an unstoppable left-footed effort from just outside the penalty area and either side of that goal he might have had more, firing just wide with one similar attempt and then hitting the base of the upright with another.
The final 20 minutes saw the home side camped in the Lincoln half and keeper Burch was forced into saves to deny Ciaran Toner, substitute Chris Dagnall and teenager Will Buckley.
The keeper was powerless though on 84 minutes as Dagnall found space on the left-hand side of the penalty area, before setting up the simplest of chances for fellow substitute McEvilly to convert from close-range.