The Edinburgh side were reduced to ten players on the stroke of the interval after Katie Fraser caught Laura Rafferty with a high boot.
And the Light Blues’ quality eventually told with the goals arriving from Katie Wilkinson twice and substitute Jane Ross.
Speaking to RangersTV after the triumph, the head coach said: “It was never really in doubt and I was pretty relaxed that we would always create something.
“I was concerned that their goalkeeper was having a worldie, which always seems to happen when teams step up and play against us.
“They were well up for the fight and we need to give Spartans a lot of credit because you could tell they were ready for the battle.
“We were well below par today and I’m not taking anything away from Spartans because I thought they made it difficult for us, but I wasn’t happy at all.
“I thought it was a real below par performance for our levels and expectations.
“It wasn’t just one or two players; it was the majority and you could see that.
“I always knew we would have that extra quality on the pitch to produce moments and score a couple of goals.
“I thought our subs helped, livening the game up and giving us a little bit more of what we needed.
“The players know we need to do a lot better than that next week.”
On the message at half-time after Fraser was shown a straight red card, Potter continued: “The message at half-time was that we needed to up our quality and move the ball quicker.
“We knew that we would create opportunities with them down to ten players, but we were making it hard for ourselves.
“We were allowing Spartans to jump out into areas they were comfortable and not shifting them enough.
“You could see that a bit more in the second half - not as much as I would have liked - but the main thing today was the three points and clean sheet.”
Wilkinson’s brace made the difference for Potter’s side as the summer arrival struck twice inside six minutes just beyond the hour-mark.
It was another couple of moments of quality from the forward, who nodded home from Chelsea Cornet’s delivery before adding her second with a deflected strike.
Potter added: “Even when we’re not playing particularly well she will pop up with a couple of goals.
“I’m sure she would have been gutted to come off before she could have had a hat-trick, but all the strikers would have been chomping at the bit to come on and make a difference.
“Both strikers came on and made a real difference in the game.
“We need to make sure we’ve got that competitive level between the squad.
“The moment of quality was the difference for us today because we weren’t quite at it.”