Boro were twice caught on the break in the second half as first Andrea Orlandi then David Lopez struck to fire the Seagulls back into the top six.
"We had to inject our own passion," he said. "In the first half the games was very slow and the ground was very quiet and sometimes that can creep into your head and into your game.
"We were comfortable in the first half but we needed more. I told then to go and create their own passion, to make their own tempo and they did that and it made a difference.
"I knew that if we could only get the first goal it would be better for us. With the players we had on the pitch it makes it very hard for Middlesbrough to score a goal."
Brighton moved up to fifth place but Poyet is not taking anything for granted.
"We are in the play-off places again but to be truthful I am not looking at the table or at other results," he said. "We have to only look after ourselves.
"We are in their now and if we win our remaining games we will stay there."