Conte's side fell behind to Laurent Depoitre's second half strike and although Marcos Alonso bagged a fortuitous equaliser, the Blues couldn't find a priceless winner as they wasted a host of chances.
Fifth placed Chelsea's attempt to salvage a troubled season by finishing in the top four is now out of their hands heading into the final game of the season at Newcastle on Sunday.
"For sure we had so many chances and you must be good to take them. It is not the first time that we are not clinical," Conte moaned.
"It is not a situation only for the strikers, but the whole team.
"Every team deserves the points they have in the table. We dropped many points. We are fifth and we must accept this."
Tottenham's 1-0 win over Newcastle on Wednesday means Chelsea can't make up the four-point gap to their third placed London rivals.
Fourth placed Liverpool are two points ahead of Chelsea and, given their vastly superior goal difference, they will condemn the Blues to the Europa League next season if they avoid defeat against Brighton at Anfield on Sunday.
"I'm realistic. This situation wasn't in our hands before this game and for sure the chances are less than before today. We have to try to do our best," Conte said.
Missing out on the Champions League would be a suitably downbeat conclusion to a lacklustre title defence from Chelsea and could prove the last straw for Blues owner Roman Abramovich as he reportedly plans to part ways with Conte.
Conte appears set to leave after just two years in charge following a contentious campaign marred by the Italian's feud with the Chelsea hierarchy over their failure to back his transfer plans.
The 48-year-old has seen a host of leading managers linked with his job in recent weeks and added fuel to the fire this week when he said his future would be revealed after the FA Cup final against Manchester United on May 19.
"This is not my task, there is a club to judge the situation and then to take the best decision," Conte said when asked if he would be in charge next term.
Huddersfield's reward for a tenacious display is another season among English football's elite after a superb result that seals their safety with a game in spare.
"If you have the passion, desire and spirit you can compete with top quality," Huddersfield manager David Wagner said.
"We have done it. I am so unbelievably proud."
In what could be Conte's final home match, he made six changes to the side that beat Champions League finalists Liverpool 1-0 on Sunday, with Olivier Giroud and Eden Hazard surprisingly left on the bench.
It was a gamble that backfired, but Conte refused to accept he made the wrong decision.
"When I make a rotation I play with players like Morata, Christensen, Willian. It is stupid to talk about the rotation," he said.
"We are Chelsea. We have a big squad. We must win."
Huddersfield took the lead with their first serious attack in the 50th minute.
When Aaron Mooy played a pass over the top of the Chelsea defence, Blues goalkeeper Willy Caballero came rushing to the edge of the area in a bid to clear.
But Depoitre beat Caballero to the ball and in the process left him sprawled on the turf, giving the Belgian striker time to net his first goal since December.
Conte responded by sending on Giroud and Hazard and Chelsea grabbed a bizarre equaliser in the 62nd minute.
Cesar Azpilicueta drilled over a low cross that prompted a panicked clearance from Mathias Zanka and the ball hit Alonso in the face before rebounding into the Huddersfield net.
Conte's side to laid siege to the Huddersfield goal.
But the visitors, fighting for survival, threw their bodies in the way time and again before Jonas Lossl superbly pushed Andreas Christensen's header onto the post.
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