"We should be delighted, but Arsenal probably deserved to get something from the game," Southgate said after the 1-1 draw at The Emirates.
"I'm incredibly proud of how they performed. We were a bit of a laughing stock last weekend and we lost a lot of pride that day."
On Mido's red-card for an attempt to bring down the ball which concluded with Gunners defender Gael Clichy being caught in the face, the Boro boss said: "I don't think he meant it.
"He caught him but he was looking at the ball the whole time and Mido is distraught because he's not a dirty player.
"It's a nasty injury, but hopefully the ref will have a look at it as I feel the lad was trying to play the ball.
"He's been to see Clichy and he's disappointed that the knock is so bad.
"Should he have his foot that high? Maybe not, but I don't think there was any intent there and a three-game ban would be harsh."