The Three Lions, who were two goals ahead in Montenegro, were pegged back through goals in the last minute of each half. The home side's second went in after Wayne Rooney was sent off for kicking defender Miodrag Dzudovic.
However, Hart insists that the side remain proud of their achievement.
"We showed true grit and true determination," he said. "I am so proud to be part of this England team.
"We are over the moon. Job done. We have got England, as a nation, to a major tournament next summer.
"We are a good bunch of players, a proud bunch of players. We played for our country knowing we had to get to a tournament."
Following his dismissal, Rooney is due to miss the opening game of next summer's finals.