The north London club had hoped to open the new arena on the site of their old White Hart Lane home last September.
But a series of delays involving safety and security measures have forced Tottenham to use Wembley as their temporary home for a second successive season.
It was feared Tottenham might have to wait until next term to move into their expensive new digs.
But on Friday they announced two further test events will take place at the end of March, which should lead to the stadium being granted a safety certificate.
The new 62 000-capacity stadium, costing a reported £850 million will then host a Premier League match against Palace on April 3 or against Brighton on the weekend of April 6 and 7.
The dates of both fixtures are subject to Brighton's progress in the FA Cup, as they will be playing in the semi-finals that weekend if they beat second-tier Millwall in the last eight.
If Brighton reach the last four, then the London derby against Palace, who could also be in the FA Cup semis, will be Tottenham's first home league game, but, if Chris Hughton's team lose in the cup, the Palace game will be put back and the Seagulls will be the maiden visitors.
"We have held discussions with the Premier League and UEFA. The following has been agreed with the Premier League, Brighton and Crystal Palace to ensure that our first game is a Premier League match prior to a UEFA Champions League quarter-final match," a Tottenham statement said.
"If Brighton win their FA Cup quarter-final match on Sunday 17th March we shall play Crystal Palace on Wednesday 3rd April and Brighton on 23rd/24th April.
"If Brighton lose their FA Cup quarter-final match on Sunday 17th March we shall play Brighton on 6th/7th April (subject to Champions League scheduling) and Crystal Palace on 23rd/24th April."
The two test events before the Premier League opening are a Tottenham Under-18 match against Southampton's youngsters on March 24 with a reduced 30 000-capacity.
The other is a Tottenham Legends game on March 30 with a reduced 45 000-capacity.
In a message to Tottenham fans, chairman Daniel Levy said: "We are delighted that we shall be able to welcome our supporters to the stadium and our players of the past and the future to take to the pitch on what will be momentous occasions for our Club.
"Please do come and see your new stadium and share these moments.
"Thank you all once again. Your support has been much appreciated during a challenging but exciting time."