The Valiants needed to win to keep alive realistic hopes of staying up and Eagles obliged with a 20-yard rocket into the top corner five minutes from time.
Their fate is still not in their own hands – even if Vale win at promotion-chasing Fleetwood on Sunday, they need relegation rivals Gillingham or Bury to slip up.
But Brown is delighted they still have a chance and hailed the impact of Eagles, who has now netted four crucial goals since arriving at Vale Park in January.
Brown said: “Chris been a great signing. He certainly helped us put the pressure on in the second half by taking Walsall on and pushing them back.
“That was the difference for us and hopefully he will do that again at the weekend.
“His goal was a wonderful strike, one of those you call a postage stamp in the top corner.
“The most important thing today was, however we played, we needed those three points going into the weekend.
“We wanted to be there at Fleetwood for a big occasion for both teams. We wanted a shot at it and we’ve got it.
“Hopefully it’s a confidence booster and the lads believe we can go and get three points again. Fleetwood are going to be very, very nervous. They know what’s at stake.
“There are still other scenarios happening on the same day so all we can do is put a good performance in and try to get that win.”
Walsall had the better chances before Eagles’ strike as Kieron Morris hit the crossbar and Vale goalkeeper Leonardo Fasan foiled James O’Connor and Amadou Bakayoko.
Saddlers boss Jon Whitney felt his young side were unlucky to lose, saying: “I couldn’t fault my lads’ effort.
“I have got five academy products in from the start. I’ve got two 17-year-olds coming off the bench and there were some great performances.
“But sometimes you get sucker-punched and kicked in the teeth and I’m gutted for the young lads that they have come away on the losing side.
“But if you are going to lose you have to give your all and they did that. I’m proud of my young players. And I’m sure my chairman, the directors and my staff are all proud of them as well.
“We have to be brave as a club because this is where we are moving forward. These players will get stronger for that experience and eventually they will turn into really accomplished players.
“Of course we are disappointed to lose. But we didn’t deserve to. It was a wonder-strike from Chris Eagles.”
Southampton have announced plans for a two-week tour of China as part of their pre-season preparations for the 2018-19 season.