Gareth Bale's powerful free-kick had given Wales the lead just before half-time, but England levelled through Jamie Vardy before fellow substitute Sturridge ended a period of strong Welsh defending with a late winner that beat goalkeeper Wayne Hennessey at his near post.
Coleman praised his players for matching England for much of the contest, and when asked if the better side won, responded: "Would you ask me that if they hadn't scored in last 30 seconds?
"We were expected to find it tough against a good team and it was, but you can't tell me that my players deserved that. Their effort and courage was fantastic.
"You have to ride your luck a bit against top teams but it's not like Hennessey pulled off save after save. So to lose like that is gut-wrenching.
"Everyone was looking at this game and you can't help but get caught up in it. You can't take the disappointment away but you can't take it forward.
"We won the first, were 1-0 up in the second at half-time and it's a bitter pill to swallow. But England showed courage to come back after conceding in the last minute to Russia."
He added: "We need to move forward as quick as we can."
Wales go into their final game sitting second in Group B with three points, and Coleman is confident his side have every chance of reaching the knockout stages by claiming a result against Russia, who drew with England and lost to Slovakia.
Coleman said: "We've watched Russia, of course. It's an exciting group because it's all on the last games.
"Russia could have got something against Slovakia, they put pressure on after starting late in the game, and Slovakia won't be an easy game for England.
"Russia will be hurting after the defeat to Slovakia but they have experience. It'll be whichever team can put the disappointment behind them from this middle game that will prevail.
"We're not asking for any favours, we've got to take care of business. We're absolutely devastated but we've got to show a bit of steel.
"We're down to the last game and we've got a chance. It's in our hands and we've got to take the chance."
Coleman had to send David Edwards on for the injured Joe Ledley in the second half against England, but said the Crystal Palace midfielder would be fit to face Russia.
"Joe had a bang, we think he'll be alright,” Coleman added. “He gave us everything, absolutely. It's a big ask to be fit for this tournament, but he's worth taking the chance with.
"I have to commend him. He was always going to run out of gas - he only had 20 minutes against Slovakia. I thought he did magnificently well."