A game played in sweaty conditions at the end of a blisteringly hot day in Philadelphia never really came to life and was decided by two goals scored in first-half stoppage time by Tosin Adarabioyo and then Delap.
Tyrique George added the third in injury time at the end of the game and the victory, achieved in front of 32,937 fans at Lincoln Financial Field, allowed Enzo Maresca’s side to bounce back from a 3-1 loss to Flamengo last Friday.
Chelsea clinched second place in Group D behind the Brazilian giants while Esperance are eliminated.
Chelsea will now travel south to Charlotte in North Carolina to face Benfica on Saturday, with the Portuguese club having earlier beaten Bayern Munich to top Group C.
That will be a first meeting of the clubs since Chelsea beat Benfica in the 2013 Europa League final, and the winner of that tie will play either Palmeiras or Botafogo of Brazil in the last eight.
Chelsea only needed a draw to advance and Maresca felt comfortable enough to make eight changes to his starting line-up following the Flamengo defeat.
Delap, Enzo Fernandez and Malo Gusto were the only players to keep their places while Josh Acheampong was among the lesser-seen faces to get a chance.
The 19-year-old right-back, making his 10th start in all competitions this season, came close with a shot saved low down by Esperance goalkeeper Bechir Ben Said on 36 minutes.
Fernandez then shot wide from the edge of the box as Chelsea controlled the first half without overly troubling the Tunisian defence until the third minute of stoppage time.
That was when Fernandez floated a free-kick into the area and Adarabioyo, up from the back, scored with a header into the bottom corner.
Esperance needed to win the game and their hopes were as good as over as Chelsea scored again two minutes later.
Fernandez, formerly of Benfica, was again the provider as he fed Delap, and the new signing from Ipswich Town controlled with his back to goal, turned and slotted a low shot into the net to get off the mark in his new colours.
With temperatures still hovering close to 30 degrees Celsius late into the evening, conditions remained far from easy for the two sides.
Both teams made changes, with defender Mamadou Sarr and Brazilian midfielder Andrey Santos getting their first Chelsea minutes since joining from Strasbourg.
Chelsea thought they had a penalty 15 minutes from time when the referee pointed to the spot as a Santos shot brushed the hand of Yassine Meriah — but the award was overturned following a long VAR check.
A third goal did finally arrive deep in injury time at the end of the game as George, another substitute, found the net with a shot that the goalkeeper should have kept out. (AFP)