Spurs offered tickets for as little as £5 for their third match in six days at the national ground, but there were just 23,926 present to see the tie settled by Dele Alli's 65th-minute goal.
The attendance was lower than for Spurs' pre-season friendly with Juventus, which attracted 26,251, but similar to the crowd at White Hart Lane for last season's League Cup third-round win over Gillingham – 26,244.
Winks told the London Evening Standard: "It's just the size of the stadium.
"It's difficult to really get that same atmosphere that you get at White Hart Lane or if you have a full stadium here.
"But the fans are always excellent and for us they were brilliant in getting us over the line."
The attendance was the lowest for a men's match at the new Wembley, excluding the FA Trophy and FA Vase finals.