England dominated from the first whistle at Marseille's Stade Velodrome on Sunday (AEST), but had to wait until the 73rd minute to take the lead courtesy of Eric Dier's thunderous free-kick.
Dier's stunning strike was no more than Roy Hodgson's team deserved for its high-tempo, all-action display, although the Three Lions would have been well ahead by that point of the game if not for a combination of poor finishing and Russian resolve.
And it was made to pay two minutes into stoppage time, when Russia captain Vasili Berezutski looped a header over Hart and into the far corner to snatch a share of the spoils.
However, Hart was determined to remain upbeat after the match, telling ITV: "It was a good performance, but a good performance in tournament football needs to be turned into a win.
"We played well, but fair play to Russia, they stayed in the game and got a goal at the end. The lad's got springs in his heels and he's looped one into the corner.
"We're going to build on that and we're going to win more games [if we play in the same way].
"There are a lot of positives to take, that's what we've got to draw on. The majority of the game was positive."
Wayne Rooney was deployed in a deeper role as Hodgson opted for a front three of Harry Kane, Raheem Sterling and Adam Lallana, and the England captain responded with an impressively assured display.
When asked for his views on the game, Rooney - who forced Igor Akinfeev into a brilliant save shortly before Dier's goal - echoed the thoughts of Hart.
"We played well. We had some good chances but didn't take them," he said.
"We got one up and conceded late on so of course we're disappointed but there's a lot of positives to take."