Two goals from Shane Long and singles from Chris Brunt and Craig Dawson at Huish Park on Tuesday night cancelled out two headers by ex-Baggie Reuben Reid to book the Premier League side's place in the third round.
Yeovil took an early lead courtesy of Reid, who featured just seven times during a frustrating spell at The Hawthorns, before Brunt and Long had the visitors ahead.
Reid equalised three minutes after half-time, before West Brom secured their place in the third round with second-half goals from Dawson and Long.
Downing said: "We knew it was going to be tough because they've started well in their league, and there were moments at 2-2 when Boaz Myhill made some very important saves to keep us in the game.
"We're delighted to go through after a very difficult game.
"Chris Brunt needed the game after coming back from an injury, and I think all the lads stepped up to the plate really well. Chris loves his shooting practices in training and he has a great left foot.
"We have a big game on Saturday (at home to Everton) so it was important for the so-called fringe players to get a game, and I think a lot of those who played pushed themselves into our thoughts for the weekend.
"We're happy with the squad and the strength in depth. After the first two games it was important to come here and avoid a banana skin. We've negotiated a few hairy moments but we're happy to be through."
Yeovil manager Gary Johnson said: "If you're going to lose you'd prefer to lose to a Premier League side but I want them to think bigger and more positively when we come up against the better teams.
"We're going to come up against better sides in League One, so this was a little test.
"It doesn't hurt to mention the things we haven't done so well. The subs weren't effective and when I make substitutions I want them to look like they've got the real will to win. But they're only young and hopefully they've learned a big lesson today.
"Some came through fantastically, but four or five didn't and they're the ones we need to drag up to the level so we can keep ourselves at the top of the league where we want to be.
"Saturday is another huge game against Doncaster. It's as big as the game tonight because it's important to get to 10 points at this stage of the season."