Their first league loss to Stoke since 1984 was their eighth of the season and came despite United being able to start Robin van Persie, Wayne Rooney and Juan Mata together for the first time.
The Old Trafford club last missed out on Champions League qualification 19 years ago.
"Each point that we drop makes it difficult for ourselves," Smalling told the club's website (www.manutd.com) on Monday.
"We can only take the next game as it comes, and other teams will drop points as well, but we need to stop dropping points. We need to put pressure on other teams.
"We've got Fulham and then Arsenal so we've got some big games. We're back at Old Trafford next week and hopefully the fans will get right behind us and that will lead us into the Arsenal game."
Although the arrival of Mata from Chelsea and the return of Rooney and Van Persie from injury briefly raised spirits at Old Trafford, United still have injury problems, with Rio Ferdinand and Nani among players still sidelined.
They also lost defenders Phil Jones and Jonny Evans during the Stoke match, with the former suffering concussion and the latter a calf problem.
"We don't really talk (about the Champions League) but we all know if anything that's the minimum we need to aim for," Smalling said.
"If we can get on a good run, we've got the players - obviously we lost a couple on Saturday, but we've got a week now until we next play so hopefully we can patch them up and get as many bodies out there as we can."
(Reporting By Josh Reich)
