An early strike from Glen Kamara was enough to secure Rangers a fourth-successive victory across all competitions and set up a semi-final clash with Hearts at Hampden Park in November.
It wasn't a vintage performance by Gerrard's side, but the former Liverpool star was pleased by how his players dug in and ultimately found a different way to win at Almondvale Stadium.
"It wasn't pretty, but I think my team have found a different way how to win, and I think we have shown this season all kinds of different sides of us," Gerrard told Rangers TV.
Gerrard knows that winning is always the most important thing, especially in a knockout cup tie, although he felt his side could have made the result more comfortable had they shown a bit more quality in the final third.
"I thought we still had the best chances and we had some opportunities on the break where, if we found that final pass, we could maybe have scored a goal to give us some breathing space," he added. "But in the main, I am delighted."
Rangers will now turn their attention back to Scottish Premiership action when they welcome Aberdeen to Ibrox on Saturday.