Leroy Sane and Sergio Aguero were also on target as City cemented their place in the Premier League's top four.
After a goalless first half, during which City felt they should have had a penalty after Fraser Forster tripped Sane, Kompany broke the deadlock in the 55th minute.
After moving back to third place in the league with Saturday's victory, City take on Arsenal in the FA Cup semi-finals and then local rivals Manchester United - two fixtures Kompany is looking forward to.
He said: "I would have been done and dusted if I needed every single time to get back in and time to adapt".
Kompany is contracted to City until 2019, but The Mirror claim that Everton and West Ham United will both be interested in signing Kompany this summer should he prove his fitness between now and the end of the season.
"For me, it's always a long journey back [after injury]". I am finally giving something back to the team.
Manuel Pellegrini guided City to the semifinal in 2015-16 - its best performance in the competition - but Guardiola said his players are still gathering the experience they need in order to conquer Europe.
"If I had to work for 15 years to have one moment like this I would do it".
With 20 minutes to go Yoshida had the golden chance to make up for allowing Kompany to rise above him and score when he found space at the far post, only to head straight down into the hands of under-fire City stopper, Bravo.
Manchester City enter this game having just beaten Hull City 3-1 but they know that Southampton will be tough opponents.
It has been a topsy-turvy start to life for Guardiola in England, his attacking football mixed with defensive deficiencies as City fight to secure Champions League qualification.
"In Spain, especially in Germany, the pressure is higher to the players, to the manager, to everybody - here it is calm", said the two-time Champions League victor. They have very good midfielders, all the strikers are so fast, they play the wingers inside and all of them have good quality.
They could register three Premier League wins in a row for the first time under Claude Puel.