Resilient Barnsley FC praised for putting brakes on West Bromwich Albion
But Bamba Diaby's own goal after 68 minutes and a header from Matheus Pereira nine minutes from time sealed a comeback and denied the Reds all three points.
Barnsley remain rooted to the foot of the table but it was a third draw in four games and second in four games against a team at the top end of the table, following their 1-1 draw with Swansea at Oakwell on Saturday.
"That's the biggest thing that has pleased me," said Murray, when asked about the character of his players.
"When the lads have suffered a lack of confidence, I've seen a real positivity around, and resilience and mentality.
"I feel sorry for the lads that we couldn't get the win but even in the 90th minute, we were still pushing."
Murray added: "I think it would have been a famous win. We had a gameplan in the way we want about the game and in the first half we executed it perfectly.
"I didn't just want us to defend, I wanted us to be on the front foot and attack them, not come here and fight for a point. I was proud and pleased of the boys."
Murray was also delighted with Woodrow's contribution.
He said: "Cauley is a player who has that in his locker - he's not just a centre-forward who pops up in the box and scores the odd goal, his all-round play is fantastic and he scores goals from all different areas."