The former Sheffield Wednesday striker, who has joined the Tykes on an initial one-month loan deal from Nottingham Forest, earned Keith Hill's men a point with a 37th-minute header.
But Tudgay remained unhappy with his performance, after allowing Charlie Austin head the Clarets into the lead five minutes into the game.
He told the club's official website: "On a personal note to get a goal I'm very pleased, but I've still got that thought in my head with my man getting a free header.
"It niggled me throughout the game and I wasn't happy for my man to score, it's not me.
"I kept going and kept working hard. I'm being harsh on myself but to let my man score is too easy."