Spurs looked set to back up Saturday's 4-0 thrashing of Manchester City with a hard-fought Europa League victory after Son Heung-min and Brennan Johnson struck in the first half to cancel out Evan N'Dicka's leveller.
Tottenham never truly gained control against Claudio Ranieri's Roma, but had the chances to kill the match in the second half after they hit the woodwork on three occasions and squandered a succession of chances.
It eventually came back to cost them as moments after Fraser Forster pulled off an outstanding save to deny Gianluca Mancini, Mats Hummels equalised in stoppage time.
The 2-2 draw has seen Spurs climb to ninth in the league phase of the Europa League, which means they would currently miss out on automatic qualification through to the last-16 stage, but Postecoglou will not shut up shop during their final three group ties.
"No," Postecoglou smiled when asked if he would ditch artistic merit for results to clinch a top-eight finish.
"Why can't it just be entertaining? Isn't it why we all come? Would you really prefer us to sketch out a couple of 0-0 draws and a 1-0 win and we get through? I don't know.
"I would much rather we're exciting to watch and maybe it is edge-of-the-seat-stuff and sometimes we don't get the reward, like we could have tonight, but no, if you want those kind of games, there are plenty in world football you can watch every weekend.
"I'd like to think when you tune into us you'll be entertained.
"There's three games to go and we're in a pretty decent position. You try to win every game you can and see where that takes you.
"I think it'll probably be clearer after the next round but it's all pretty tight from what I understand. That's what you expect."
Spurs head to Rangers in a fortnight with the Scottish Premiership club one place higher in the league table in eighth after a 4-1 win at Nice.
A similar scoreline could have occurred in north London despite an even first 45 where Son's fifth-minute spot-kick, after Hummels brought down Pape Sarr, was cancelled out by N'Dicka's header from Paulo Dybala's free-kick.
Brennan Johnson's ninth goal of the campaign in the 33rd minute from Dejan Kulusevski's cut back - not long after Stephan El Shaarawy had a volley ruled out for offside - seemed to put Tottenham in the ascendancy.
However, Roma had two more efforts disallowed for offside at the start of the second half via Manu Kone and Artem Dovbyk before Angelino struck the crossbar with a rasping volley.
Spurs did regroup and after Pedro Porro clipped the crossbar, Dominic Solanke headed against the woodwork from Archie Gray's cross to set up a nervy finale, which reached a dramatic conclusion as Forster's flying save to deny Mancini counted for little when Hummels fired in from Angelino's cross.
Postecoglou added: "Yeah, we should have killed it off a lot earlier.
"At 2-1 you're always keeping the opposition in the game and they're throwing men forward, so disappointing we weren't able to see the game out. Ultimately we're still in a decent position."
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