Robin van Persie had given the champions a half-time lead, but Lallana's goal rescued a point, which was well deserved, according to their manager.
"I want to congratulate my players, it was an amazing effort and they fully deserve the credit today," Pochettino told BBC Sport,/em>
"It's not easy to come to a ground like Manchester United's and get a positive result, but we prepared for the match thinking we could win.
"We're aware that Manchester United are an amazing club, but we came here to win."
The result means that Southampton have now taken four points from trips to Anfield and Old Trafford, following their 1-0 triumph at Liverpool last month.