Roberto Mancini's men continue to lack the dynamism which propelled them to the Barclays Premier League title last season but, after their Champions League frustrations, winning was all that mattered this weekend.
City, Mancini and Kompany all came in for criticism after a loss at Ajax last Wednesday that left them facing an uphill battle to maintain their interest in Europe's elite competition.
The team were again far from their best as the Swans visited the Etihad Stadium, but a fine Carlos Tevez strike settled an otherwise dour game 1-0 and secured a fourth Premier League win in succession.
Kompany told City TV: "We won the game and we move on now. We are doing all right in the league.
"The most important thing here would always have been the result, certainly after Wednesday. We got the result.
"I don't think it will be one of the games we will remember forever but it does put us in the position where it is comfortable enough for us to go into the West Ham game.
"You couldn't do much more than this week. It was about winning and we did that."
City are also moving on off the field having confirmed the appointment of former Barcelona sporting director Txiki Begiristain as their new director of football overnight.
The 48-year-old will work alongside manager Mancini in "first-team recruitment and operations" as part of the club's plan to establish themselves at elite level.
Brian Marwood, the club's football administrator, will take up a new role of managing director of the new City Football Academy.
The CFA, being built at a reported cost of £200million on a site adjacent to the Etihad Stadium, will be a state-of-the-art training complex for around 400 players.
Their football on Saturday did not initially match their lofty ambitions.
Swansea forward Michu had the only serious chance of an insipid first half but he was denied by a brave save from Joe Hart.
But having been criticised for altering his formations too much in Amsterdam, Mancini made a good tactical switch at half-time.
Aleksandar Kolarov, who had been mostly ineffective on the wing anyway, was withdrawn with a foot injury and Mancini changed the shape by sending on forward Mario Balotelli.
City began to look more cutting but, even so, it still needed another good save by Hart from Michu to keep it goalless before Tevez struck on the hour.
Kompany said: "Swansea have always been a good team. It was never going to be an easy game and I think, for us, it was good to keep the clean sheet and create chances.
"The most important thing is we are now on track for the next game."
City remain unbeaten in the Premier League and have now kept successive clean sheets on the domestic front, their only two of the season.
They will now hope to take advantage of a week without a midweek fixture to prepare for next week's trip to Upton Park.
Mancini will also want some good news on the injury front, with defender Micah Richards facing assessment after leaving the field on a stretcher and wearing an oxygen mask.
Richards, in only his fourth game back after an ankle injury, went down in agony clutching his knee in the closing stages of a match which had 12 minutes injury time.
Kolarov will also be assessed while Pablo Zabaleta, Maicon, Jack Rodwell and Javi Garcia will hope to get nearer to fitness.
Swansea, who nullified City in the first half and gave them some anxious moments after the break, will be in action in midweek in the Capital One Cup.
In what will be a big week for the Welsh side, the Swans will travel to Anfield in the knockout competition on Wednesday before hosting Chelsea in the Premier League next Saturday.
Manager Michael Laudrup will need to make at least two changes for Anfield after goalkeeper Michel Vorm was also carried off with a groin injury and Angel Rangel incurred a fifth yellow card of the season.
Laudrup said: "I have to see how people are for Wednesday, think about that game and the Chelsea game. But I can guarantee it will be a competitive team.
"If we win at Liverpool then we are in the quarter-finals and that would be the first time in the story of Swansea. That would be very good for the club.
"They are another two big games this week but, if we play like this, I am sure there will be something waiting for us in these two games.
"I only want to only see the positive things here because there are so many."