Jackson missed with three successive second half penalties and then failed to convert Andrew Trimble's late try. Those misses proved costly and Ulster will get overtaken by the winners of the clash between Leinster Rugby and Glasgow Warriors tonight.
Ulster dominated the first half of the season in their domestic league, winning13 straight games before Munster took their unbeaten record at the end of 2012. Since then, they have lost four and drawn one of their last eight PRO12 matches.
As well as fighting for a place in the PRO12 Play-Offs, Ulster have to travel to Twickenham Stadium in a fortnight's time to face top English club Saracens in the Heineken Cup quarter-finals.
Anscombe had warned his players ahead of the game in Edinburgh not to think too far ahead, but now the pressure has been turned up on the Ulster camp after another sloppy defeat.
"It is easy to take the eye off things. We have worked hard this season to be in the position we currently are. But if the mind wanders to London in a few weeks time, then we are going to end up losing all three games. Where will our season be then? What will all the hard work have been for?" Anscombe told his players.
"We have come through a tough period and we had players missing for various reasons. We're not blaming that - we are where we are now and we must now push on.
"There are players come back in again next week. The focus for everyone involved is to realise that they are playing for their positions. If they want to be involved against Saracens then they have to show me why over the next two weeks.
"Guys are playing for their places, it is as simple as that. That is their only focus - if they want to play at Twickenham, then they must put their hand up!"