The Football Association's decision for the manager to start in the new year raised eyebrows, just as interim Lee Carsley's penultimate selection did for Thursday's must-win match.
England captain Harry Kane was dropped to the bench on a night when anything but victory would see Greece win Group B2 having secured a historic 2-1 triumph at Wembley.
Carsley's bold selection backfired last month but worked this time as the Euro 2024 runners-up silenced a packed Olympic Stadium, where Ollie Watkins justified his selection with an early opener.
Jordan Pickford produced some important saves before Jude Bellingham saw a low shot come back off the post and deflect in off Greece goalkeeper Odysseas Vlachodimos.
England knew a win by two goals would see them take control of Group B2 thanks to a superior head-to-head record with the hosts, but Jones wrapped up a fine victory with a beautiful finish.
The triumph in Athens means victory against the Republic of Ireland in Carsley's final match before handing over to Tuchel will see them promoted.
Meanwhile, France and Italy have qualified for the quarter-final stage of the revamped Nations League.
France drew 0-0 with Israel in a match requiring a high-security operation in and around Paris, a week after violence erupted in Amsterdam in connection with Israeli club Maccabi Tel Aviv's visit for a Europa League match.
Around 150 Israel supporters were among a crowd of less than 20,000 inside the 80,000-capacity Stade de France and there was a brief scuffle between some fans at one point in the first half.
The draw was enough to secure France a top-two finish in their group along with Italy, who beat Belgium 1-0 away thanks to Sandro Tonali's 11th-minute goal.
They advance to the quarter-finals in March, from which the lineup of the Final Four mini-tournament in June will be determined. Germany and Spain were the first two nations to qualify for the quarter-finals in October.
Prolific Manchester City striker Erling Haaland was again among the scorers for Norway in their 4-1 win in Slovenia, while North Macedonia beat Latvia 1-0.
The Republic of Ireland beat Finland thanks to Evan Ferguson's fourth senior international goal.
The Irish also had Caoimhin Kelleher to thank after the Liverpool goalkeeper saved a second-half penalty to get them across the line.
with AP
