I got it. The trick for me is to keep my left thumb really low on the back of the neck - a bit like when I'm playing barre chords, but maybe even lower. Then, I can anchor my index finger on the 1st fret and play all the notes legato. Just have to get it up to speed now....

BTW, on the same song, I can't play the solo exactly because my guitar only has 21 frets, not 22. I get round it by, e.g. playing a double pick at 20, then full step bend at 20, so it stays in key. But is it much different playing a 22 fret model compared with a 21? I'm thinking about upgrading later this year, but really not sure whether to stay with the reliable strat-type or go with something different.