Keane has urged the England international to remain grounded after he took his tally to five in seven starts for Arsenal across all competitions.
"It's took him a while to get settled, hasn't it?" Keane told ITV Sport.
"Listen, he's had a good week, the guy needs to relax, try playing over the next seven, eight or nine months when the real big crunch games come and you're on about looking at certain players for Arsenal.
"Tonight was a lovely game for Arsenal, when they have that much possession they are great to watch, they could have scored seven, eight, nine goals easily.
"But the crunch games are coming up over the next few months when the cold nights come on, we'll see maybe half the Arsenal players with gloves on.
"So I wouldn't be getting carried away with Walcott. He's got a lot to do. Lovely goals tonight, simplicity, his movement, finishing – but he's got to do it over the next year or two."