Shaw, the youngest England player to start a World Cup game since Michael Owen, showed no sign of nerves going forward, but he gave away a free-kick in a dangerous position.
Borges stepped up and bent a dipping shot over the wall and onto the bar via the fingertips of the athletic Foster.
The only other point of note in the dull first period came just before the half hour when a Ross Barkley touched James Milner's cross into the path of Sturridge. The Liverpool striker shaped to shoot, but Duarte brought him down with an out-stretched leg.
Referee Djamel Haimoudi waved play on, much to Sturridge's dismay.
Fortune favoured England in a move straight after the break. Shaw's shot looped straight into the path of Sturridge in the box, but the striker mis-controlled, allowing Costa Rica goalkeeper Keylor Navas to smother the ball.
Giancarlo Gonzalez dumped Barkley on the turf with a wrestling move, but Frank Lampard, appearing possibly for the last time in an England shirt, wasted the free-kick, driving straight into the wall.
Phil Jones, a few minutes after failing to control a simple pass, Jones then gave possession away inside his own half, but luckily for England Christian Bolanos shot straight at Foster.
Milner played a lovely one-two with Sturridge to set the striker free inside the box, but he curled an inch wide.
Hodgson tried to inject some life into his team by introducing Raheem Sterling.
The England support then demanded Wayne Rooney enter the pitch. Instead they got Steven Gerrard, who was cheered by the fans, who suspected this might be the captain's final England appearance.
Rooney finally came on with 15 minutes left.
The Manchester United striker attempted an audacious chip over Navas within minutes of his introduction. He also came within inches of turning home Lampard's cross in the dying minutes.
England kept on trying, but they could not beat Costa Rica - a team ranked below Scotland by FIFA.
The locals and the Costa Ricans inside the stadium joined forces to belittle England at the end, shouting "eliminado".
You do not have to be a multi-linguist to work out what that means.
And you do not have to be a genius to work out that this has been a sorry World Cup for England.