After Alberto Paloschi had opened the scoring after 19 minutes at White Hart Lane, Swansea defended resiliently with Fabianski making a string of saves to deny Christian Eriksen, Harry Kane and Eric Dier.
The former Arsenal goalkeeper's performance was in vain, though, as goals from Nacer Chadli and Danny Rose secured a 2-1 victory for Tottenham, leaving Swansea three points clear of the bottom three in 16th.
READ MORE | 'Pochettino: Tottenham showed they have stomach for title fight'
"Lukasz was outstanding he made some great saves," Curtis told reporters. "He had to make some saves but some of them were world class.
"He had a fantastic game."
Fabianski was not the only one to come in for praise from Curtis, who is hoping Swansea can build on their performance in the coming weeks and safeguard their top-flight status.
He added: "There is some sort of solace from the performance in that we defended well but this is a difficult place to come to.
"They're a terrific side, the pace, the power and the technical ability that they have got [make] it a really tough place to come.
"They kept driving at us all the time and you sensed from the crowd that they could do it.
"It disappointing for us but we have got to show same spirit again for the rest of the season."