City had their three-point advantage at the top of the Barclays Premier League wiped out as Darron Gibson's deflected strike condemned them to a 1-0 defeat.
Kompany, who had been making his return from a four-match suspension, told City TV: "In a season you are going to lose games - unfortunately you have to lose games at some point. We didn't want it to happen but we look forward and we still want to go on and bring the league back to Manchester City."
With rivals Manchester United beating Stoke on the same night, City now lead their neighbours only on goal difference at the summit. The result rounded off a frustrating January for City which began with a defeat at Sunderland on New Year's Day and also saw them go out of both domestic cup competitions.
Manager Roberto Mancini, who remarkably blamed his own complacency for the loss, will hope for better fortune in February, starting with Saturday's home clash against Fulham.
City had long spells of possession at Goodison Park but Everton's makeshift centre-back pairing of Tony Hibbert and John Heitinga coped well with the challenge.
Gibson struck, via a deflection off Gareth Barry, on the hour and City could not muster a reply, although they felt they could have had a penalty when an Aleksandar Kolarov cross struck Phil Neville's arm.
Kompany said: "I thought probably losing was a bit harsh but we knew it was going to be difficult. We are just disappointed we couldn't be more dangerous after dominating the game like this.
"In the end it was a deflected goal - there was nothing the goalkeeper or anyone could do about that. We never felt like we were in too much danger but I guess it is their strength - they contained us pretty well and when they had the chance, with a bit of luck, they scored.
"From that moment it was always difficult. I think it was one of those kind of games where if you don't have a bit of luck with you on a day like this you don't win."