It was uncertain whether Argentina winger Lavezzi would play but PSG coach Laurent Blanc included him in the starting lineup and he scored his fourth league goal of the season from close range in the 18th minute following Ibrahimovic's deflected shot.
The Sweden striker made it 2-0 five minutes after the break with his 16th league goal at the Parc des Princes this term before Kagelmacher scored in his own net two minutes later.
The capital club now have 58 points from 25 games ahead of second-placed Monaco, who visit mid-table Bastia on Saturday.
PSG rested several first-choice players ahead of Tuesday's Champions League last 16 match at Bayer Leverkusen.
The changes did not stop PSG dominating third-from-bottom Valenciennes, who could have gone behind in the fourth minute but for centre back Kagelmacher clearing Ibrahimovic's effort off the line.
The hosts had several chances to get on the scoresheet before Lavezzi, who netted only three league goals last season and had been in poor form recently, deservedly put PSG ahead before wasting two more opportunities soon after.
Ibrahimovic eventually grabbed the second goal with a half-volley after the ball fell perfectly for him following an effort by new signing Yohan Cabaye. PSG sealed the game seconds later when Kagelmacher diverted Lavezzi's low cross in his own net.