England trailed France 16-3 at the start of the second quarter after two tries for wing Yoann Huget.
But Mike Brown’s first international try four minutes before the break sparked a stunning revival. England scored 18 unanswered points, including a try on his debut for Luther Burrell, to lead 21-16 after 56 minutes.
Maxime Machenaud and Alex Goode exchanged penalties, and it looked like England had done enough to secure a winning start to their Six Nations campaign before Gael Fickou came off the bench to score a try with two minutes remaining and snatch the victory for les Bleus.
“After falling behind in the first minute, we were up against it and it was always going to be tough from there," Leicester Tigers prop Cole told the Leicester Mercury.
"But the way we took the game to France to get back in it and score some tries was credit to the guys.
"They say you make your own luck but a couple of bad bounces cost us dearly in the opening 15 minutes. In the end, they cost us the game.
"It would have been easy to fold when two tries went against us with unlucky bounces – but we didn't.
"This team is not going to quit. We will play until the game is over and we finally got some reward for the pressure we were creating when Mike Brown scored.
"Then Danny Care nearly scored but was stopped on the line and we took the lead with our second try. We didn't die. We kept coming back at them.
"I was gutted they nicked it at the end after we had fought back into the game. We had shown some real guts and then a guy comes off the bench and does what he is there for."
Cole is among four Tigers players who reported back to England’s training base at Pennyhill Park on Sunday evening to prepare for Saturday’s match against Scotland at Murrayfield.
Tom Youngs, used as a second-half replacement in Paris, is joined by Ed Slater and Ben Youngs in the 33-man squad at this stage. Freddie Burns returns following his two appearances for England Saxons, while Stephen Myler (Northampton Saints), Rob Webber (Bath) and Richard Wigglesworth (Saracens) remain with their Aviva Premiership clubs.
Adam Thompstone's debut for the England Saxons ended in a 16-16 draw with Scotland 'A' at Scotstoun on Friday.
Leicester Tigers wing Thompstone was one of four players making their Saxons debut in Glasgow with Semesa Rokoduguni, Kyle Eastmond and Dan Robson drafted into their backline.
And they had helped the Saxons to a 13-6 lead early in the second half before Scotland A fought back to eventually claim a draw through Tom Heathcote’s late penalty.
Saxons fly-half Burns and opposite number Henry Pyrgos exchanged early penalties before England scrum-half Robson was sin-binned for an alleged stamp.
A Pyrgos penalty gave Scotland the lead for the first time before England hit back with a penalty try from a scrum, converted by Burns, to go in 10-6 ahead at the break.
Another penalty put the Saxons further ahead before Burns was replaced by Henry Slade, who became the second England player to receive a yellow card when he was caught offside.
Byron McGuigan took advantage to score Scotland’s try, converted by Heathcote, to tie the scores.
Elliot Daly’s 63rd-minute penalty looked to be enough to give the Saxons victory before Heathcote’s late leveller.
Tigers youngsters Henry Purdy, Harry Thacker and Will Owen, meanwhile, kicked off their Under-20s Six Nations campaign with England in a 21-15 defeat to France.
Purdy and Thacker started the game at the Stade Leo Lagrange in Draguignan while Owen was among the replacements.
France played the final 17 minutes with 14 men after Anthony Rochet received a red card. They led 16-10 at that stage and Thacker scored a try for England with eight minutes remaining as they threatened a comebck.
However, Jean-Baptiste Singer scored a crucial late try to overcame France's numerical disadvantage and seal the win.
England had raced into a 7-0 lead through Tom Stephenson’s try, converted by Billy Burns. Two penalties from Baptiste Serin - either side of a yellow card for Thacker - trimmed the gap to just one point and that was how it stayed until half-time.
Early in the second-half Serin kicked France ahead for the first time but Burns replied before home centre Xavier Mignot and England winger Henry Purdy were both sent to the sin bin on 54 minutes.
Almost immediately, France took a six-point lead thanks to a try from No.8 Francois Cros and Serin's conversion before Rochet’s red card and the late drama.