Andrew Trimble and Tommy Bowe went over to help last year's beaten finalists to a 13-6 lead at the break with Ryan Lamb kicking two penalties for the hosts.
But Ulster kicked on in the second half with Jared Payne andRaboDirect PRO12 ambassador Dan Tuohy crossing to seal a bonus-point victory at Franklin's Gardens, and Anscombe felt the province sent a message out to their Heineken Cup rivals.
"That was a bit of a statement," said Anscombe. "We weren't even thinking about the bonus point. We were more than happy to come here and take the win.
"The way Rory Best, John Afoa and Tom Court handled the front row was outstanding.
"You saw that they [Northampton] changed their front row after about 60 minutes because of the work that had been done on them.
"We knew if we didn't get that in our front row that we would be in trouble."
In Pool Four's other match, Glasgow Warriors fell to a third straight Heineken Cup defeat as Castres boosted their qualification hopes.
The sides went in level at the interval with Warriors fly-half Scott Wight and Castres scrum-half Rory Kockott kicking two penalties apiece in a tight opening period.
However Wight missed a penalty soon after the restart and Kockott took advantage to kick his side 9-6 ahead.
Gregor Townsend's men did pick up a bonus point though, leaving them seven behind Castres who sit in second place.
Fetu'u Vainikolo's early try helped Connacht to a first Heineken Cup win over French opposition as they saw off Biarritz in Galway.
Connacht dealt with back row Eoin McKeon's sin-binning just before the interval, with Tiernan O'Halloran's late yellow card failing to trip them up as they went second in Pool Threebehind Harlequins with a 22-14 victory.
Dan Parks added the extras and landed a drop goal after the Tongan winger went over but Dimitri Yachvili kept Biarritz in it with three penalties.
But Parks kicked two penalties of his own and another drop goal to stretch the province's lead, though Biarritz back row Imanol Harinordoquy got a late try which Yachvili failed to convert for a bonus point.