The Warriors went into the match as heavy underdogs and trailed to a first half Max Lowe goal at half time before earning a replay.
Despite being behind, McMenamin was happy with his side’s performance and told the players they would need to take their chance when it came – which Mark McGuigan did with a diving header in the 70th minute.
“We rode our luck a bit in the second half,” said McMenamin post-match.
“Aberdeen had a good couple of chances, but the boys worked hard and limited them especially in the first half where I don’t think they had many shots at goal.
“In the second half they come out at us and we manage to weather the storm before scoring a cracking goal.
“Hopefully this result can galvanise the team after a difficult season.
“We had a couple of boys coming into the team today, big Andy Munro was making his debut and he really made a difference – he was excellent.
“We have had a difficult first half of the season, things have changed, but it’s all about building on it now and we have a big game at home to Airdrieonians to concentrate on now.
“Even after we scored I think we had a couple of chances to sneak a win but we would have taken a draw before kick off, we’re delighted with it and I’m sure the board will remind me how delighted they are with it later.
“We were working on crossing and finishing on Thursday night.
“I told them you’re not going to get a lot of chances against a team like Aberdeen and if you get a chance from eight yards you need to take it and, fortunately, big Mark gets across the defender and finishes with a good header.
“Everybody to a man worked as hard as they could and I’m delighted for them.
“Their confidence was knocked a wee bit after two very difficult games and they come to Pittodrie with everyone expecting us to take a hiding.
“I’m delighted with the way they worked and they deserved a draw.”
Before the goal Aberdeen were pushing hard for a second but were denied more than once by Graeme Smith in the Stenny goal who was in fine form.
“Buzz is a top goal keeper who has played at a very good level,” McMenamin added.
“We missed him last week but he wasn’t going to stay out for this game and I thought he was excellent.”
The replay will take place on Tuesday, January 29, at Ochilview with the winner facing either Dundee or Queen of the South at home in the fifth round.
“We’ve got to try and make it difficult for them.”
“They won’t be too used to playing on astroturf but at the same time they will be able to pass the ball quick on a nice surface
“I’ve been thinking about things all week and how to get a positive result, we had 300 odd fans on Saturday which was excellent and I’m really pleased for them.
“Just getting this tie has helped us financially and let us get a few players in this window and hopefully the replay strengthens the pot further and lets me get another couple in.”