The hometown hero might have expected an easier opening encounter but found the big serving German a tricky player to see off.
The Swiss struggled to convert a number of break points in the opening set before Becker finally gifted him a break with a double fault.
Becker then went a break up at 3-2 in the second set but that seemed to finally kick Federer into gear as he broken straight back to love before rattling off the three games he needed to win the match.
Federer has won the indoor tournament in five of the last six years – only losing in the final to Novak Djokovic in 2009.
He will next face Thomaz Bellucci or Go Soeda in the second round.
Juan Martin Del Potro, Richard Gasquet and Stanislas Wawrinka are amongst the other big names playing in the event.
Meanwhile, in Valencia the stand-out result came from former world number one Lleyton Hewitt who dumped out fourth seed Juan Monaco 6-3 6-4.
Monaco went into the week with an outside chance of making the ATP World Tour Finals but those hopes now appear to be in tatters after this latest loss.
ATP Basel results
3-Richard Gasquet (France) beat Robin Haase (Netherlands) 4-6 6-3 6-2
1-Roger Federer (Switzerland) beat Benjamin Becker (Germany) 7-5 6-3
Julien Benneteau (France) beat Andrey Kuznetsov (Russia) 4-6 6-3 6-3
Lukasz Kubot (Poland) beat Lukas Lacko (Slovakia) 6-4 6-4
ATP Valencia results
Albert Ramos (Spain) beat Rajeev Ram (U.S.) 6-3 6-3
5-John Isner (U.S.) beat Fabio Fognini (Italy) 6-3 6-7(4) 7-5
Ivan Dodig (Croatia) beat Philipp Kohlschreiber (Germany) 6-2 7-6(6)
Fernando Verdasco (Spain) beat Tommy Robredo (Spain) 6-3 6-4
Lleyton Hewitt (Australia) beat 4-Juan Monaco (Argentina) 6-3 6-4
David Goffin (Belgium) beat Pablo Andujar (Spain) 6-2 6-2
- Sports & Recreation
- Juan Monaco