Keane played for his country against Brazil in midweek, and Mowbray said: "I was half tempted to put him on the bench today after midweek but I started him and he got his goals.
"He showed against Brazil the talented boy he is - he's scored a few goals for us - and he showed it again today, taking the two opportunities which came along. Hopefully we can keep him fit."
Mowbray described his team's performance as "composed" and "professional".
He added: "We knew it would be a difficult day. They've given us problems in the past and the pitch probably wasn't going to suit what we were trying to do but we had a bit of composure and professionalism."
Falkirk counterpart Steven Pressley said: "I thought we controlled the game for long periods and I'm devastated for the players. I thought they gave so much to the game.
"We had a number of opportunities over the course of the 90 minutes and created by far the better chances.
"The pitch was an issue for part of game but our level of football, considering the pitch, was brilliant today.
"The second goal was avoidable and disappointing but the performance was excellent and I've no complaints in terms of the players' effort."
The former Celtic player believes the Bairns can beat relegation, despite being three points adrift at the bottom of the table.
He added: "My belief is we'll stay in this division. We've got St Johnstone here next week and winning that game becomes our focus."
Source: Team Talk
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