City were a long way from their best but O'Reilly stabbed home in the 84th minute before Kovacic made sure in time added on to put City four points clear of sixth-placed Chelsea.
James Tarkowski had struck the post for Everton in the first half.
Brighton's hopes of European football suffered a blow as Bryan Mbeumo scored twice in a dramatic 4-2 win for Brentford in west London.
Mbeumo gave Brentford a ninth minute lead, but Danny Welbeck headed Brighton level in first-half stoppage time.
Three minutes into the second half, Mbeumo restored Brentford's advantage, and it was 3-1 10 minutes later when Mbeumo turned provider for Yoane Wissa, whose strike deflected off Jan Paul van Hecke.
Things got worse for Brighton when Joao Pedro was sent off for lashing out at Nathan Collins, and although Kaoru Mitoma pulled one back with nine minutes to go, Christian Norgaard struck in stoppage time to wrap up the win.
Lesley Ugochukwu's stoppage-time equaliser made sure Southampton cannot do worse than Derby's record low Premier League points total as they snatched a 1-1 draw at West Ham.
Jarrod Bowen had fired the Hammers in front two minutes into the second half at the London Stadium, but Saints got a deserved point late on when Chelsea loanee Ugochukwu found a way through a crowded penalty area.
Bournemouth achieved a club record Premier League points tally as they reached 49, but it should have been more as they played out a 0-0 draw at 10-man Crystal Palace.
Palace's Chris Richards was sent off in first-half stoppage time, accounting for two of the nine yellow cards shown by referee Sam Barrott in a tetchy encounter.
