Burnley looked like they would pay for playing the last 10 minutes a man light following David Edgar's red card when David Norris scored with four minutes remaining.
But Carlisle still found time to head home a corner and give Brian Laws' side a draw.
Ipswich boss Roy Keane - who was a team-mate of Laws at Nottingham Forest in the early 1990s - is keen to add to his squad before the end of the transfer window and needs to find a cutting edge if his side are to mount a challenge this year.
Keane did hand Spurs winger Andros Townsend an instant debut after arriving on loan on Thursday but captain Jon Walters was not in the squad as speculation continues to link him with a move to Stoke.
Ipswich forward Tamas Priskin fired over in a lively start before Townsend ended a mazy dribble with a shot which he dragged wide.
Both sides were struggling to get their passing game in order but it was the visitors who came close to an opener in the 25th minute.
Captain Graham Alexander and Dean Marney combined well in midfield to release the overlapping Ross Wallace down the left flank.
Wallace surged into the area before picking out Wade Elliott on the stretch but his shot was brilliantly tipped over the crossbar by Fulop.
Wallace and Townsend traded long-range strikes as the half drew to a disappointing close.
Keane made a change at the break, with midfielder Grant Leadbitter replaced by Colin Healy.
Healy was into the action quickly as he did well to block a Marney strike on the edge of the area as the Clarets slowly took control of the midfield battle.
Elliott did well to buy a yard of space on the right and from his deep cross Martin Paterson was able to flick a header narrowly wide.
Ipswich were fired into life and Jon Stead saw his low cross-shot hacked off the line by Edgar - who then blotted his copybook by getting sent off.
Edgar had been booked in the first period by referee Grant Hegley and saw red with 11 minutes left for a trip on Norris.
And Norris finally broke the deadlock with four minutes left.
Healy delivered a deep corner from the right and when Stead guided a header down into the danger area, Norris was able to flick in from in front of Brian Jensen for his third goal of the week.
But the Clarets stole a point in stoppage time when Carlisle thundered in a header from Alexander's corner.