Luke Varney gave Championship side Leeds a 15th-minute lead in the fourth-round tie and Ross McCormack made it 2-0 five minutes after the interval.
Spurs, who were without striker Jermain Defoe due to a slightly twisted hip, pulled a goal back on 58 minutes but were unable to find an equaliser.
"Leeds made the most of their opportunities," Villas-Boas said. "It was good to come back into the game but we maybe could have done a little better.
"Leeds were really up for it. We had some opportunities to go 2-2, but maybe not too clear-cut.
"Our second half was definitely better than the first, but we could have done better not to go 2-0 down.
"Unfortunately today we missed Defoe, who always gives us an option, through injury, but we'll have him back for the Norwich game."