Bergin made it eight goals in four matches as Spurs overcame an early jolt to eventually progress comfortably into the next round of a competition they were runners-up in last season.
Enfield, who play a division below Spurs in the FA Women's Premier League South-East, took the lead from a corner in the 20th minute.
However, Rosie Kmita scored a cracker to equalise and settle any nerves two minutes later and Spurs took over from there at Cheshunt FC.
Mollie Kmita set-up Bergin for 2-1 before the break and the forward - called into NI's squad for the Euro 2017 qualifier in Georgia later this month - made it 3-1 on the hour, following up after Leah Rawle struck the post.
Shannon Moloney completed the scoring after a scramble from Jenna Schillaci's corner in the 75th minute.
"We had a bit of a shock going 1-0 down but recovered well and got the job done," said manager Karen Hills.
"The most pleasing thing was we were able to give time to some new players. Barbara Holman did well in goal, Ronnell Humes, who has progressed through the ranks and is part of our development scheme at Barnet & Southgate College, came on in the second half and worked really hard and Ellie Butterwick also came off the bench.
"All in all, a good performance and after beating Plymouth in the League Cup, it's been a good week for us so far."
The week continues with a trip to high-flying Cardiff in the FAWPL South on Sunday (2pm).
Spurs Ladies v Enfield: Barbara Holman, Mollie Kmita (Ellie Butterwick 77), Alex Keown, Jenna Schillaci, Lucia Leon, Amy Cooper, Shannon Moloney, Riana Soobadoo, Avilla Bergin (Jenny Moore 78), Rosie Kmita (Ronnell Humes 65), Leah Rawle. Substitutes (unused): Toni-Anne Wayne, Katie O'Leary.