A Pool Four shock looked likely when Glasgow raced into a 15-0 lead courtesy of tries by flanker Josh Strauss and wing Sean Lamont.
But Northampton recovered to score three tries in six minutes either side of half-time - George Pisi (2) and Stephen Myler the beneficiaries - before wing Vasily Artemyev's late score secured a 24-15 result and an unlikely winning bonus point.
"It could have gone a different way, and what you cannot do is underestimate sides in the Heineken Cup. We didn't do that - Glasgow are a quality team," Mallinder said.
"They have a big heavy pack with a back-row that gets around the field and backs that can take their chances.
"We were under a bit of pressure, but what was pleasing was the way we came back and kept our composure. In the second half, we controlled the game.
"I think we are in decent shape. When we are on top of our game we are a hard side to beat, and we need to keep at that intensity every single week."
Saints now have just a five-day turnaround before tackling Castres in Toulouse next Friday night, with Glasgow in the same boat before hosting Ulster.
Mallinder added: "I don't think it is quite right that Castres have got seven days to recover from their game two days ago, and we've got just five and having to travel to France as well, but that is the way it is.
"It is always a massive challenge playing in France, but if we play at the top of our game then we've got a good chance of winning."
Mallinder admitted that memories were evoked of last season's poor European campaign when Glasgow took the early initiative.
"The Scarlets came here last season and turned us over, which started our Heineken Cup season badly, and we never really recovered," he said.
"But we didn't panic today and there were positive messages out there on the pitch."
Myler and Ryan Lamb each booted a conversion for Northampton, but Myler also missed four shots at goal that could easily have proved costly.
Glasgow, though, did not even gain the consolation of a losing bonus point, which they will feel should have been the minimum return from their journey south.
"We are disappointed to come away with nothing," Glasgow head coach Gregor Townsend said. "We played very well for 30 minutes, then let them back in to the game.
"And we were disappointed in the end to not get a losing bonus. It was very frustrating.
"The two tries at the end of the first half got them back into the game. I felt we could have been further ahead, but it looked like a momentum shift when they scored those tries."
Glasgow skipper Al Kellock added: "When you break the game down, we were fantastic for the first 30 minutes and then we made silly mistakes that led to two tries.
"But it's about moving forward and making sure we learn from it.
"We had the ability to win today and that is credit to the squad we have built. We have got belief.
"We are in a tough pool, there is no doubt about it, and you need to be able to bounce back quickly in this tournament."
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