Playing in its first Olympic competition since 1960, Britain took the lead on 19 minutes after Senegal failed to clear a free kick into the box from captain Ryan Giggs. Fellow Welshman Craig Bellamy pounced on the loose ball, driving a shot into the ground that bounced to the left of keeper Ousmane Mane.
But Britain, which has been playing together for less than three weeks, was unable to convert its first half possession into chances against a muscular Senegalese side that showed attacking intent throughout. Bellamy and Giggs, playing in his home stadium of Old Trafford, looked dangerous on occasion, but the team as a whole lacked control.
It was fortunate to go into halftime with the lead. Six minutes after Bellamy's strike, British keeper Jack Butland fluffed a clearance into the path of Sadio Mane. But the striker, who had looked Senegal's most dangerous player of the evening, chipped high and wide.
The Senegalese, most of whom play in the European leagues, took the attack to Britain after the break.
Ibrahima Balde almost equalized two minutes into the half with a low shot, but Butland got a hand to it and turned the ball past the post. The keeper again had to save well in the 55th, tipping over a shot from Saliou Ciss. From the corner that followed, defender Danny Rose cleared a header by Papa Gueye off the line.
Konate's goal came in the 81st, as Britain was hoping to hold out for a scrappy victory. The striker, who plays for Israel's Maccabi Tel-Aviv, ran on to a through ball that exposed Britain's defence fragility before chipping Butland from close range.
A frantic final few minutes saw Britain substitute Marvin Sordell hit the bar from close range. Konate could have scored a second, but his shot went wide.
"The fact that we got our noses in front and didn't go on to nail the game was a frustration," said Britain coach Stuart Pearce. "We've got another tough game Sunday, the group looks like it might be wide open."
Britain next plays United Arab Emirates, which lost against Uruguay earlier Thursday, on Sunday at Wembley Stadium.
Britain felt it should have had a penalty before the equalizer after Ciss clattered into Bellamy on the edge of the box.
"You could see that it was inside the box and that's why I was so angry," said Neil Taylor. "You could end someone's career in a flash like that."
The match was played in front of a near capacity crowd at Old Trafford, most of whom had never seen a British football team take the field in a major competition.
The country's English, Welsh, Scottish and Northern Irish football associations had previously resisted the idea of a Britain team, fearing it could jeopardize their independence within global football. Given that Britain is hosting the games, they agreed to field a team this time around, though only Welsh and English players are represented.
Fresh attention was drawn to the issue when the match program incorrectly listed Welshman Joe Allen as English.
Critics on Twitter noticed that the Welsh players, Giggs included, did not appear to be singing the national anthem ahead of the match.
Coach Stuart Pearce said he wasn't bothered about that.
"Some enjoy singing, some don't," he said. "It's fine with me."