Rangers had earlier beaten St Mirren by the same scoreline to briefly pull level with their Glasgow rivals on 78 points.
Forrest was drafted in for a rare start after coming off the bench to score in last weekend's Scottish Cup semi-final win over Aberdeen.
The Scotland international volleyed home from the edge of the box on the half-hour mark and started and finished the move that doubled the lead midway through the second half.
Celtic substitute Adam Idah sliced the ball into his own net as he tried to clear Antonio Portales's shot in the 74th minute.
That goal set up a nervy finish for the travelling fans but Dundee's hopes ended when substitute Michael Mellon headed wide from a deep free-kick in stoppage time.
"We've played better at times than what we did today but it was all-important to get the three points," said Celtic manager Brendan Rodgers.
"Two great finishes (by James Forrest). He's one of the fathers of the great success that this club has had over the last decade. Very humble, very understated but what he has, he's got immense quality."
Champions Celtic have taken 13 points from a potential 15 since their defeat to Hearts in early March.
They are strong favourites to win the title with just four rounds of fixtures to play and the Old Firm still to meet at Celtic's own ground.
Earlier, Rangers took the lead in the 32nd minute when St Mirren defender James Bolton put the ball into his own net attempting a goalline clearance.
Home striker Mikael Mandron deservedly levelled with a well-placed header five minutes later and the second half was a struggle for the visitors before Cyriel Dessers popped up with a powerful header in the 74th minute to seal the points.
Manager Philippe Clement, whose side had been winless in three league games, admitted the victory was not "sexy" but said the points were all that mattered.
"They did what they need to do and that is winning games," he said. "That is the first thing.
"Was it our most sexy game? No. But it is also the circumstances and all respect to St Mirren also."
