Keane wasted two gilt-edged chances to cement his hero status among Celtic fans, thousands of whom turned out to welcome him to Parkhead at midnight.
And he was ultimately upstaged by fellow new boy Chris Maguire, who sealed a 1-0 win for Killie shortly after half-time.
Mowbray said: "For me, it's about trying to win every game - we've said that.
"We haven't won tonight, so we're obviously disappointed, but we keep going.
"It's a long way to go.
"I'm sure there'll be some twists and turns and we'll look to try to get the next three points when they come."
He added: "When those big games come along, let's see how the flow of the points total is.
"You go away and win at Ibrox, and you win at Easter Road, and you win at Parkhead in both games and let's see when the pressure comes on where the points go and how the table's looking."
Asked how the low of Tuesday night compared with the high of Keane's unveiling, Mowbray snapped back: "Highs for who? Highs for you or highs for me?
"Players coming in, that's what we're trying to do. I've been saying that for a long time.
"We going to try to build this club to the point where we can compete very successfully, both domestically and in Europe."
But the Celtic boss denied the circus around Keane has been an unwelcome distraction.
"I don't think so; I thought he played well.
"He brings a different dimension to our team.
"At times, he made the team tick by positions he takes up on the pitch.
"I'm sure he'll be a huge benefit for us as we move on."
Of the result, he praised Kilmarnock's resilience.
"We came here trying to claw back the three points and it didn't go our way," he said,
"You've got to give them a lot of credit - I thought they defended fantastically well.
"But disappointed with the nature of the goal.
"At the other end, we did create chances; we ended up in some decent areas but didn't take them.
"That's football - you have to sometimes take it on the chin and keep going, keep building up and look forward to the weekend and try to get it going."
Mowbray lost Andreas Hinkel and Jos Hooiveld to injuries that look set to keep them out of Sunday's Active Nation Scottish Cup fifth-round tie at Dunfermline.
The one bright spot was Scott Brown coming off the bench for his first appearance for more than three months.
The Scotland midfielder took the armband and looks set to become Mowbray's permanent captain.
Asked if he was happy with his first SPL win since taking charge last month, Killie boss Jimmy Calderwood said: "That is an understatement.
"It looked as if the script was written.
"First 10, 15 minutes, Robbie was getting on the ball and causing us massive problems.
"We had to hang on but we did.
"We started to get into the game, young Maguire playing up front for us, keeping the ball up and he's always dangerous on the counter-attack.
"Good football for the goal and then Celtic threw everything at us.
"They even put Scotty Brown on at left-back.
"Our keeper's made a couple of great saves and we rode our luck."
Calderwood believes coming from behind to draw 4-4 with Dundee United on Saturday helped inspire Tuesday night's performance, as did Keane's arrival at Parkhead.
He said: "With Robbie signing, it brought a big Celtic crowd out and the atmosphere was fantastic, and it lifted the boys."