Wasps might have gone the same way - as they did in their pool game at Edinburgh last season - had replacement fly-half Alex King not come on to close out a win set up by Paul Sackey's first-half try, 11 points from the boot of Jeremy Staunton and a late penalty by Mark van Gisbergen.
They dug in successfully and Castres were unable to add to prop Akventsi Giorgadze's 45th-minute try and a conversion and two penalties from former Scotland fly-half Gordon Ross.
McGeechan said: "We know there are things we want to do better and it was ugly, but we stuck in there.
"In the first half we got the control we needed, but in the second the conditions deteriorated and we didn't have that control.
"We had to play for 25 minutes in our own 22 and apart from one moment - and the players aren't sure about their try - we held up well.
"We didn't fear it (another Edinburgh) because our discipline was good and in the last 10 minutes we didn't give the initiative back to them. In previous games, we might have done that."
The victory sets up the prospect of an epic clash next Saturday away at Perpignan, who opened their campaign by claiming a bonus point from a 25-10 win against Pool One makeweights Treviso.
Wasps may have to go there without Josh Lewsey, who limped off after 55 minutes with a hamstring injury that will have worried England coach Andy Robinson as he prepares to name a 30-strong squad for the autumn internationals this week.
Robinson and McGeechan can take heart, though, from another successful showing by Phil Vickery, the World Cup winning England prop who played his first full 80 minutes since a back operation last winter.
Vickery said: "We can't go beating ourselves up about this performance - this was a game we had to win and now we have to look ahead to Perpignan.
"You have to keep a sense of perspective. It's European rugby, you play different teams with different refs and it takes time to adapt.
"It doesn't come much harder than Perpignan away but you live for games like that. It'll be a battleground - a real step up - and we'll have to be very streetwise."