Derek McInnes was also ordered off for pushing Brown, who is working for free for the administration-hit club, after 17-year-old substitute McGhee headed Hearts 2-1 ahead in the 88th minute following an error from Nicky Weaver.
Referee Alan Muir also dismissed Hearts left-back Kevin McHattie for a challenge on Calvin Zola midway through the second half, which allowed Niall McGinn to convert a penalty and cancel out Jamie Walker's 17th-minute opener.
Locke felt the passion on display at Tynecastle was a good advert for the Scottish Premiership as Hearts knocked the seventh point off their 15-point penalty.
Locke said: "The technical areas aren't exactly a million miles apart. You think you are a bit harshly done by and then you score a winning goal. Anybody that doesn't show a bit of passion, questions would be asked.
"I didn't think Billy did a lot wrong. His momentum took him into their technical area. I didn't think he could move that quickly. It's something out of nothing. We will go and have a beer with Derek now and it will be forgotten.
"The game was action-packed and I think we should be talking about how good a game it was with two teams who were desperate to get three points. There was a lot of quality on both sides and I think we should be talking about that rather than Basher Brown."
McInnes had not spoken to Brown by the time he spoke to the press.
"I'm disappointed," the Dons boss said. "I have never seen a member of staff come into the opposition's technical area to celebrate a goal the way he did.
"But as soon as I put my arms on him to try and push him back, I am guilty. I have apologised to the referee. I am disappointed in myself but I was surprised at the reaction."
Both managers were reluctant to give definitive views on two other talking points - the penalty and another incident which saw Peter Pawlett booked for diving when Walker challenged him in the box - as they had not seen the replays.
Locke, who felt the second incident was a penalty, disagreed with the spot-kick that was given.
"I would need to see it again but from where I was I felt it was harsh, I felt he got the ball," he said. "But at Partick I thought it was a harsh penalty and when I saw it again I thought Bobby Madden was spot-on."
McInnes, who rued his side's wasted chances in the first half and their defending at both goals, said: "I think McHattie has put his arms round him. For me it looked a penalty kick and the right decision.
"Peter felt he was caught. My initial reaction from real speed was it was a penalty. I know he has got a reputation that has gone before him at times and maybe this is a case of this today.
"I do feel Peter Pawlett was outstanding and has been since the start of the season. I will be able to make a better comment on that when I have seen it."
Locke praised everyone at the club for their commitment after they cut the deficit at the bottom of the league to nine points.
"It was a fantastic effort from everyone," he said. "We go down to 10 men and you think if it's a point it will be a great result.
"But it's a good ball in and big Jordan peeled round to the back stick and it's a great header. He is 17 years old but I had no qualms about putting him on, he is a fantastic football player and he's going to have a big future."