Walcott gave the Gunners the lead in the first half and Robin van Persie, making his first return to the Emirates Stadium since joining United last summer, claimed the champions a point when he scored from the penalty spot before half-time.
Walcott said: "There are still three very important games left. We have just got to concentrate on ourselves.
"We can't keep relying on teams to beat whoever. As long as we get the three points and don't drop any points that's all we can do."
The England winger was happy to take a point from a game which the Gunners dominated before United came back into it.
"I think if you look back both (teams) will think it was fair," Walcott added on Sky Sports 1.
"Both teams didn't really want to lose the game. Overall it will be a good point come the end of the season.
"We have got to be proud of ourselves, it was a good point."