Sam Szmodics put the visitors ahead in the 38th minute, with Jimenez levelling from the spot midway through the second period.
Liam Delap restored Ipswich's lead with another penalty but Jimenez had the final say in stoppage time.
The result leaves Ipswich third from bottom of the table after falling agonisingly short of recording two consecutive league wins for the first time this season.
Mid-table Fulham enjoyed the lion's share of possession in the first half but Ipswich broke the deadlock against the run of play with their first chance of note.
Delap surged forward and fed Leif Davis, whose cross was headed onto the bar by overlapping right-back Ben Johnson. After a scramble to clear the danger, Szmodics was on hand to crash the ball off Calvin Bassey and past Bernd Leno.
It took until the 69th minute for the Cottagers to get back on level terms. Harry Wilson found a pocket of space once more following Rodrigo Muniz's pass and was brought down inside the box by Sam Morsy.
Referee Darren Bond initially said no penalty but changed his original decision after a VAR check and Jimenez stepped up to slot his spot-kick to Christian Walton's left.
Delap put Ipswich back ahead just two minutes later after he was fouled by Timothy Castagne and they went close to a third when substitute Jack Clarke struck a post.
But Fulham continued to probe for an equaliser and it arrived in the first minute of stoppage time.
Jimenez picked himself up after being fouled by Davis and coolly found the top-right corner to secure his side a share of the spoils.
"When you lead twice you always feel a little bit disappointed not to win," Ipswich boss Kieran McKenna told Sky Sports.
"I'm so proud of the way we played. We had some chances to get a two-goal margin but weren't able to take them."
Fulham boss Marco Silva praised the performance of Jimenez, who now has eight Premier League goals this season.
"He showed the character and the calmness," he said. "There are many positives I take from the game. In some moments we cannot concede like we did and we have been punished by some mistakes.
"Everybody can talk about European hopes. I prefer our team to speak on the pitch. If you want to be fighting you have to be much more ruthless and aggressive."
