T&TEC started quickly and built a three-goal lead through Neaves, Bailey and Teesdale, but goals from Jesse Alexis (40th) and ex-Defence Force striker Odelle Armstrong (61st) saw St Francois pull back to 3-2 early in the second half.
However, the Super League leaders were able to pull away again when Bevon Bass (75th) and Teesdale both scored within four minutes.
In other results on Sunday, Tobago's FC Phoenix beat Club Sando 3-0 and M&M Harvard edged Stokely Vale, another Tobago club, 2-1.
National Super League Results:
Sunday
T&TEC 5 (Kevon Neaves 2nd, Joel Bailey 8th, Sylvester Teasdale 37th, 79th, Bevon Bass 75th) St.Francois Nationals 2 (Jesse Alexis 40th, Odelle Armstrong 61st)
1976 FC Phoenix 3 (Dominique Kerr 71st, 88th, Sean Christmas 83rd) Club Sando 0
M&M Harvard 2 (Glen Walker 65th, Christon Freitas 85th) Stokely Vale FC 1 (Darron Williams 78th)
Upcoming matches:
Today
M&M Harvard vs Club Sando, Dibe Recreation Ground, 4 p.m.
Sunday
1976 FC Phoenix vs WASA FC, 3.30 p.m., Canaan, Bon Accord
M&M Harvard vs Club Sando, 3.30 p.m., Dibe Recreation Ground
Stokely Vale FC vs T&TEC, 3.30 p.m., Plymouth Recreation Ground
Joe Public FC vs Club Sando, 3.30 p.m., Marvin Lee Stadium
Defence Force vs St. Francois Nationals, 3.30 p.m., TBA
Valtrin FC vs QPCC, 3.30 p.m., NUGFW Ground, Valsayn
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