The Fir Park outfit are now 11 points clear of the automatic relegation berth with just four games remaining but Alexander insists that has not been his focus.
He said: "I know the league says we are 11 clear of Hamilton but our ambition is to take as many points as we can in the last four games and finish as high as we can.
"That's a big win for our tally and results elsewhere have been reasonably good, but I don't want that to impact our approach in our next games.
"We have to commit to every game and make sure we take more points. Staying up was the first target but our effort and commitment shouldn't stop there so we will keep our foot to the floor."
The defeat saw St Mirren drop to eighth in the table and manager Jim Goodwin felt only poor finishing had marred an otherwise strong performance.
He said: "I'm scratching my head a little bit trying to work out how we've lost that game.
"I thought for the majority of it the players were excellent. They've completely dominated the game in terms of possession – we must have had about 70 per cent of the ball.
"It's probably the best we've played for a number of weeks in terms of building the game and we looked really tidy on the ball. But the afternoon sums up our season in terms of not being able to kill teams off."
St Mirren midfielder Ryan Flynn limped off with a hamstring injury late in the first half and Goodwin feared he may now lose him for most of the remainder of the season.
He added: "Flynny has been out of the team for a long time and sometimes that happens. The body isn't used to that intensity.
"He's overstretched for a ball and his hamstring has gone. So that will be two to three weeks.
"I thought he was having a really good game as well. It's a bitter blow for Ryan after waiting so patiently to get into the team."