Mowbray's men will escape the east end pressures next week when they travel across Glasgow to Ibrox, with the chance to close the gap on SPL leaders Rangers to four points.
However, asked about the anxiousness which envelopes Celtic Park at times, Celtic manager Mowbray replied: "That's been there since day one, has it not?
"The team didn't win the league last year and there is an anxiety in the stadium but the players have to deal with that.
"We have found on several occasions this year that once you score the second goal, the anxiety disappears, the crowd enjoy the event and everything loosens off.
"(Yesterday), we didn't manage to get that second goal and so the anxiety remained.
"But at the end of the day we won the match and got the points."
Keane, who stoked the fires ahead of the trip to Govan by claiming the pressure was now on Rangers - whose game at St Johnstone was frozen off - claims tension always plays a part when chances go begging.
"Sometimes that happens in games," he said after yesterday's game.
"At 1-0, you are always liable to give the other team a chance but, all in all, we dominated the game and didn't look like losing it.
"But it is important in these kinds of situations to get the second goal.
"I have been lucky enough to play at big clubs before and the pressure comes with that.
"You are playing for a club that demands success and, as players, it is up to us to deliver it.
"We are well aware that people expect us to win these kind of games but sometimes they are not as easy as people think."
Keane took his tally to three in five appearances and the Republic of Ireland striker, named man of the match against United, believes he and his Celtic team-mates are starting to hit top form.
"It is starting to gel together," he said.
"People are getting to know each other's games.
"I said when I first came that it would take me three or four games to get in to the swing of things and I feel I am starting to do that.
"I'm starting to score now and long may it continue."
The only consolation United manager Peter Houston could take from a lacklustre display was the fact Hibernian failed to stretch their one-point lead in third place by losing at Motherwell.
Houston said: "Motherwell are on a fantastic run just now so it brings the three teams closer.
"We have two home games coming up, starting with Falkirk next week and they had a fantastic result against Aberdeen.
"We have to play the way we did at the Falkirk Stadium recently when we hammered them 4-1, but we have let it slip at Tannadice recently.
"So for the next couple of weeks, we will be at the players to turn Tannadice in to a wee bit of a fortress."