Euell scored one goal and laid on the other as the Saints cruised to a surprisingly easy victory.
But for the heroics of City's brilliant Brazilian goalkeeper Adriano Basso the league leaders would have been on the end of a hammering.
Southampton dominated almost from the first kick and could have been 3-0 in front even before Stern John's scrappy 35th minute opener.
John should have opened the scoring after just four minutes when he was expertly picked out by a cross from Mario Licka, but headed hopelessly wide from four yards.
Basso produced a flying save to keep out Indigo Idiakez's curling free-kick soon after.
John wasted another golden chance in the 11th minute when he brilliantly controlled a Gregory Vignal centre, but dragged his shot wide.
It was Idiakez's turn to miss a hitter in the 20th minute when he put a free header into the side-netting from Jermaine Wright's cross.
Southampton finally took the lead ten minutes before half-time when John was on hand to hammer into the roof of the net after Euell's cross had been headed across goal by Bradley Wright-Phillips.
City managed only one shot on target in a strangely one-sided first half, prompting manager Gary Johjnson to call from the bench Steve Brooker and Ivan Sproule for the start of the second period.
The changes made little difference to the flow of the game and Southampton were only denied a greater victory by Basso's exertions.
Basso produced a stunning one-handed save to keep out Licka's close-range volley in the 69th minute.
City's man of the match then pulled off two breathtaking stops in the space of a few seconds to deny Wright-Phillips.
First Basso saved at the feet of Wright-Phillips, who looked certain to score, and then tipped the former Manchester City striker's long-range shot wide.
Euell eventually put the game beyond City when he tapped home from close range after Basso could only parry another Wright-Phillips shot.