The midfielder earned his 15th cap as the Three Lions recovered from going 2-1 down in the second half to take maximum points.
Everything looked rosy at half-time in Kiev as England led 1-0 through Andy Carroll's header in the 23rd minute.
Scott came closest to scoring before that with a piledriver half-volley from 25 yards that Swedish goalkeeper Andreas Isaksson did well to claw out of the top corner.
It all changed at the start of the second half as two quick goals from former Villa defender Olof Mellberg put the Swedes 2-1 up.
However, England hit back again with goals from Theo Walcott and Danny Welbeck to secure a valuable win.
The result leaves England second in Group D, level on points with leaders France, who top the group on goal difference after their 2-0 win against co-hosts Ukraine.
Sweden are out of the running and it means England need a draw or win against Ukraine in their final group game on Tuesday to progress.
There is no Spurs interest tonight as the tournament starts its final round of group matches - Russia take on Greece and Czech Republic face Poland in Group A.
The next Spurs player in action is Rafa van der Vaart, whose Holland side need to beat Portugal and hope results go their way with Germany facing Denmark in Group B on Sunday.
All final group matches kick-off at 7.45pm UK time.