Jake Cooper gave the hosts the lead, heading home an Oliver Norwood cross.
But Hull levelled shortly afterwards as influential midfielder Tom Huddlestone put Abel Hernandez through to fire low into the centre of goal.
Huddlestone was again the provider as he found Robertson, who struck home from outside the box for a vital goal.
It was Hull's first victory away from home since February and one which guaranteed them a top-six finish with three games remaining.
Bruce's men were top of the Championship in February before a poor run of two wins in nine matches saw them slip down the table.
But their hopes of securing automatic promotion are still alive as fourth-placed Tigers - eight points adrift of Brighton in third and second-placed Burnley - have a game in hand over their rivals.
Reading's form continues to tumble as the season nears its conclusion with this their fourth successive league defeat.
In front of 12,949 fans, Reading's lowest of Championship season, the hosts started the brighter.
With the score at 1-1, Reading thought they had won a 58th-minute penalty when referee Fred Graham pointed to the spot after Harry Maguire appeared to have fouled Deniss Rakels.
But Hull's players persuaded Graham to consult with an assistant and the referee then changed his mind, awarding a corner instead.
Reading continued to pile on the pressure, but it was Hull who took the three points as Robertson popped up from left-back to clinch the winner.
Reading manager Brian McDermott told BBC Radio Berkshire:
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"We need a break, we've lost four on the bounce now, it's a horrible little run we're on.
"All we can do is dig in for the next three days, it's hard being a football manager at times like this.
"It's tough on all of us but we'll come through this adversity and be stronger as a result as this.
"We started the game really brightly but at 2-1 up Hull had nous in their team to see the game out, maybe we need a bit of nous."
Hull manager Steve Bruce:
"We got off to a horrible start but the response was good. Overall, the game could have gone either way.
"It was a typical Championship match but it's gone our way. It was a much better performance from us."
Andrew Robertson's late winner secured victory for Hull over Reading as Steve Bruce's men confirmed at least a place in the Championship play-offs.