The defender scored after 16 minutes following a foul by County centre-half Declan Drysdale on Bruce Anderson.
The visitors dominated large spells in the second half but Alex and Dominic Samuel both wasted good headers, while Blair Spittal had a low drive saved by Max Stryjek.
Martindale felt his side should have done much better going forward but was delighted to see them claim another clean sheet at the other end.
He said: "I think Ross County dominated large spells of possession in the second half but I thought we had the better chances.
"We defended our box really well in the second half, Max had a save and the only other one was the header.
"I was reading a stat and that's 22 clean sheets from 55 games. At a club our size, and I mean this in the nicest possible way, I think that's pretty phenomenal, pretty phenomenal that you've got a goalkeeper that's doing that.
"I remember when we had Liam Kelly it was similar. That comes from the unit that's in front of you.
"It's a foundation to work on and that's the objective to work on.
"It's cup football, the objective before the game was to make sure you're in the hat for the next round."
County manager Malky Mackay made six changes to the team for the trip to West Lothian and admitted he was down to the bare bones through injury, with five of his seven outfield players on the bench not fit enough to come on.
He said: "The team was absolutely no disrespect to the Scottish Cup or Livingston. What we played was what we had left.
"Six of the nine on the bench were not fit to come on. We had two outfield players left, Jordan White and Harry Paton, and they both came on.
"We took some heavy hits on Tuesday and had a suspension.
"We had 14 we could play, so I filled the bench with the rest to make it look as if it was viable.
"It was no excuse - because we were beaten fair and square."