Harper coolly finished off a Stephen Hunt cross in the dying seconds to give the Royals their first away win of the season.
Royals boss Steve Coppell said: "It was a great result for us. Having not won away from home all season we had to approach the game in a certain manner and our game-plan went reasonably well.
"We were mindful of the 120 minutes Middlesbrough had played in midweek and we knew it was going to be factor in the last 10 to 15 minutes and that proved to be the case.
''A clean sheet would have been significant and to get all three points was a massive, massive boost. Psychologically I hope it is is going to have a real long-term effect.''
Coppell added: "There are an awful lot of managers who are uncomfortable down at the bottom and that discomfort continues for ourselves. But we have that determination both individually and as a team.''
Southgate said: "The way they play is full of energy and we knew we'd be lacking some of that today. But it's disappointing because we should have seen it through and we got caught with a sucker punch.
"We tried to freshen the team up a little bit but it was going to be one of those under-par days. Reading did a good job because they played in the way they knew would make us use as much energy as we had left.''